COMMUNICATIONS I N M AT H E M AT I C S · Jan 'ustek, Department of Mathematics, Faculty of Science, The University of Ostrava, 30.dubna 22, 70103 Ostrava 1, Czech Republic, jan.sustek

VOLUME23/2015No. 1

ISSN 1804-1388(Print)

ISSN 2336-1298(Online)

C O M M U N I C AT I O N SI N M AT H E M AT I C S

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Olga Rossi, The University of Ostrava & La Trobe University, Melbourne

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Ilka Agricola, Philipps-Universität Marburg

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Anthony Bloch, University of Michigan

George Bluman, The University of British Columbia, Vancouver

Karl Dilcher, Dalhousie University, Halifax

Stephen Glasby, University of Western Australia

Yong-Xin Guo, Eastern Liaoning University, Dandong

Haizhong Li, Tsinghua University, Beijing

Vilém Novák, The University of Ostrava

Geoff Prince, La Trobe University, Melbourne

Gennadi Sardanashvily, M. V. Lomonosov Moscow State University

Thomas Vetterlein, Johannes Kepler University Linz

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Lukáš Novotný, The University of Ostrava

Jan Šustek, The University of Ostrava

Available online athttp://cm.osu.cz

Editor-in-Chief

Olga Rossi, Department of Mathematics, Faculty of Science, The University of Ostrava,30. dubna 22, 701 03 Ostrava 1, Czech Republic &Department of Mathematics and Statistics, La Trobe University Melbourne, Victoria3086, Australia, olga.rossi a©osu.cz

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Ilka Agricola, FB 12, Mathematik und Informatik, Philipps-Universität Marburg, Hans--Meerwein-Str., Campus Lahnberge, 35032 Marburg, Germany,agricola a©mathematik.uni-marburg.deAttila Bérczes, Institute of Mathematics, University of Debrecen, P.O. Box 12, H-4010Debrecen, Hungary, berczesa a©science.unideb.huAnthony Bloch, Department of Mathematics, University of Michigan, 2074 East Hall, 530Church Street, Ann Arbor, MI 48109-1043, USA, abloch a©umich.eduGeorge Bluman, Department of Mathematics, The University of British Columbia, 1984Mathematics Road, Vancouver, B.C. V6T 1Z2, Canada, bluman a©math.ubc.caKarl Dilcher, Department of Mathematics and Statistics, Dalhousie University, Halifax,Nova Scotia, B3H 4R2, Canada, dilcher a©mathstat.dal.caStephen Glasby, Centre for Mathematics of Symmetry and Computation, School of Math-ematics and Statistics, University of Western Australia, 35 Stirling Highway, Crawley6009, Australia, stephen.glasby a©uwa.edu.auYong-Xin Guo, Eastern Liaoning University, No. 116 Linjiang Back Street, Zhen’an Dis-trict, Dandong, Liaoning, 118001, China, yxguo a©lnu.edu.cnHaizhong Li, Department of Mathematical Sciences, Tsinghua University, Beijing, 100084,China, hli a©math.tsinghua.edu.cnVilém Novák, Department of Mathematics, Faculty of Science, The University of Ostrava,30. dubna 22, 701 03 Ostrava 1, Czech Republic, vilem.novak a©osu.czGeoff Prince, Department of Mathematics and Statistics, La Trobe University Melbourne,Victoria 3086, Australia, G.Prince a©latrobe.edu.auGennadi Sardanashvily, Department of Theoretical Physics, Physics Faculty, M. V. Lomo-nosov Moscow State University, 119991 Moscow, Russia, sardanashvi a©phys.msu.ruThomas Vetterlein, Department of Knowledge-Based Mathematical Systems, Faculty ofEngineering and Natural Sciences, Johannes Kepler University Linz, Altenberger Straße 69,4040 Linz, Austria, thomas.vetterlein a©jku.at

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Communications in Mathematics 23 (2015) 1–11Copyright c© 2015 The University of Ostrava 1

A note on normal generation and generation of groups

Andreas Thom

Abstract. In this note we study sets of normal generators of finitely pre-sented residually p-finite groups. We show that if an infinite, finitely pre-sented, residually p-finite group G is normally generated by g1, . . . , gk withorder n1, . . . , nk ∈ 1, 2, . . . ∪ ∞, then

β(2)1 (G) ≤ k − 1−

k∑i=1

1

ni,

where β(2)1 (G) denotes the first `2-Betti number of G. We also show that

any k-generated group with β(2)1 (G) ≥ k − 1 − ε must have girth greater

than or equal 1/ε.

1 IntroductionIn the first part of this note we want to prove estimates of the number of normalgenerators of a discrete group in terms of its first `2-Betti number. It is well-knownthat if a non-trivial discrete group is generated by k elements, then

β(2)1 (G) ≤ k − 1. (1)

The proof of this statement is essentially trivial using the obvious Morse inequality.The following conjecture was first formulated in [13].

Conjecture 1. Let G be a torsionfree discrete group. If G is normally generatedby elements g1, . . . , gk, then

β(2)1 (G) ≤ k − 1.

If G is finitely presented, residually p-finite for some prime p, then Conjecture 1,i.e., the inequality β(2)

1 (G) ≤ k−1 is known to be true, see Remark 2. In this note,we give a proof of a variation of this conjecture, which also applies to the non-torsionfree case. In Theorem 7 we show the following result

2010 MSC: 16S34, 46L10, 46L50Key words: group rings, `2-invariants, residually p-finite groups, normal generation

2 Andreas Thom

Theorem 7. Let G be an infinite, finitely presented, residually p-finite group forsome prime p. If G is normally generated by a subgroup Λ, then

β(2)1 (G) ≤ β(2)

1 (Λ).

In particular, if G is normally generated by elements g1, . . . , gk ∈ G of ordern1, . . . , nk ∈ 1, 2, . . . ∪ ∞, then

β(2)1 (G) ≤ k − 1−

k∑i=1

1

ni.

The proof is based on some elementary calculations with cocycles on G tak-ing values in C[G/H], for H ⊂ G a normal subgroup of finite index, and Lück’sApproximation Theorem [10].

In Section 5 we prove in Theorem 8 that if a k-generated group G satisfiesβ

(2)1 (G) ≥ k−1−ε, then the shortest relation in terms of the generators must have

length at least 1/ε. A theorem of Pichot [15] already implied that the girth of theCayley graph of G with respect to the natural generating set becomes larger andlarger if ε is getting smaller and smaller. Our main result is a quantitative estimatethat implies this qualitative result. We prove in Theorem 8:

Theorem 8. Let G be a finitely generated group with generating set S. Then,

girth(G,S) ≥ 1

k − 1− β(2)1 (G)

.

The main tool is an explicit uncertainty principle for the von Neumann dimen-sion.

2 Residually p-finite groupsIn this section we want to recall some basic results on the class of residually p-finitegroups and show that various natural classes of groups are contained in this classof groups. Let us first recall some definitions.

Definition 1. Let p be a prime number. A group G is said to be residually p-finite,if for every non-trivial element g ∈ G, there exists a normal subgroup H ⊂ G ofp-power index such that g 6∈ H. A group G is called virtually residually p-finite ifit admits a residually p-finite subgroup of finite index.

The following result relates residually p-finiteness to residual nilpotence andgives a large class of examples of groups which are residually p-finite.

Theorem 2 (Gruenberg). Let G be a finitely generated group. If G is torsionfreeand residually nilpotent, then it is residually p-finite for any prime p.

Another source of residually p-finite groups is a result by Platonov, see [16],which says that any finitely generated linear group is virtually residually p-finitefor almost all primes p. In [1], Aschenbrenner-Friedl showed that the same is true

A note on normal generation and generation of groups 3

for fundamental groups of 3-manifolds. Gilbert Baumslag showed [3] that any one-relator group where the relator is a p-power is residually p-finite. For any group,its image in the pro-p completion is residually nilpotent.

We denote the group ring of G with coefficients in a ring R by RG. Its elementsare formal finite linear combinations of the form

∑g agg with ag ∈ R. The natural

multiplication on G extends to RG. The natural homomorphism ε : RG→ R givenby

ε

(∑g∈G

agg

):=∑g∈G

ag

is called augmentation. We denote by ωR the kernel of ε : RG → R; the so-calledaugmentation ideal.

In the proof of our main result, we will use the following characterization offinite p-groups that was obtained by Karl Gruenberg, see [7] and also [8], [5], willplay an important role.

Theorem 3 (Gruenberg). Let G be a finite group and let ωZ ⊂ ZG be the aug-mentation ideal. The group G is of prime-power order if and only if

∞⋂n=1

ωn = 0.

3 `2-invariants of groups3.1 Some definitions

`2-invariants of fundamental groups of compact aspherical manifolds where intro-duced by Atiyah in [2]. A definition which works for all discrete groups was givenby Cheeger-Gromov in [4]. Later, a more algebraic framework was presented byLück in [10]. We want to stick to this more algebraic approach.

Let G be a group and denote by CG the complex group ring. Note that thering CG comes with a natural involution f 7→ f∗ which is given by the formula(∑

g∈Gagg

)∗=∑g∈G

agg−1.

We denote by τ : CG→ C the natural trace on CG, given by the formula

τ

(∑g∈G

agg

)= ae.

It satisfies τ(f∗f) ≥ 0 for all f ∈ CG and the associated GNS-representationis just the Hilbert space `2G with orthonormal basis δg | g ∈ G on which G(and hence CG) acts via the left-regular representation. More explicitly, thereexists a unitary representation λ : G → U(`2G) and λ(g)δh = δgh. Similar to theleft-regular representation, there is a right-regular representation ρ : G→ U(`2G),given by the formula ρ(g)δh = δhg−1 .

4 Andreas Thom

The group von Neumann algebra of a group is defined as

LG := B(`2G)ρ(G) = T ∈ B(`2G) | ρ(g)T = Tρ(g),∀g ∈ G.

It is obvious that λ(CG) ⊂ LG, in fact it is dense in the topology of pointwiseconvergence on `2G. Recall that the trace τ extends to a positive and faithfultrace on LG via the formula

τ(a) = 〈aδe, δe〉.

For each ρ(G)-invariant closed subspace K ⊂ `2G, we denote by pK the orthogonalprojection onto K. It is easily seen that pK ∈ LG. We set dimGK := τ(pK) ∈[0, 1]. The quantity dimGK is called Murray-von Neumann dimension of K. Lückproved that there is a natural dimension function

dimLG : LG-modules → [0,∞]

satisfying various natural properties, see [10], such that dimLGK = dimGK forevery ρ(G)-invariant subspace of `2G.

We can now set

β(2)1 (Γ) := dimLΓH

1(Γ, LΓ),

where the group on the right side is the algebraic group homology of Γ with co-efficients in the left ZΓ-module LΓ. Since the cohomology group inherits a rightLΓ-module structure a dimension can be defined.

Remark 1. The usual definition of `2-Betti numbers uses the group homologyrather than the cohomology. Also, usually `2G is used instead of LG. That thevarious definitions coincide was shown in [14].

3.2 Lück’s Approximation Theorem

A striking result, due to Lück, states that for a finitely presented and residuallyfinite group, the first `2-Betti number is a normalized limit of ordinary Betti num-bers for a chain of subgroups of finite index, see [10] for a proof. The result saysmore precisely:

Theorem 4 (Lück). Let G be a residually finite and finitely presented group. Let

· · · ⊂ Hn+1 ⊂ Hn ⊂ · · · ⊂ G

be a chain of finite index normal subgroups such that⋂∞n=1Hn = e. Then,

β(2)1 (G) = lim

n→∞

rk((Hn)ab)

[G : Hn]= limn→∞

dimC H1(G,Z[G/Hn])⊗Z C

[G : Hn].

This result has numerous applications and extensions, we call it Lück’s Approx-imation Theorem.


3.3 Lower bounds on the first `2-Betti number

It is well known that the first `2-Betti number of a finitely generated group G isbounded from above by the number of generators of the group minus one. A morecareful count reveals that a generator of order n counts only 1 − 1

n . Similarly,lower bounds can be found in terms of the order of the imposed relations in somepresentation. More precesily, we find:

Theorem 5. Let G be an infinite countable discrete group. Assume that thereexist subgroups G1, . . . , Gn, such that

G = 〈G1, . . . , Gn | rw11 , . . . , rwkk , . . . 〉,

for elements r1, . . . , rk ∈ G1 ∗ · · · ∗Gn and positive integers w1, . . . , wk. We assumethat the presentation is irredundant in the sense that rli 6= e ∈ G, for 1 < l < wiand 1 ≤ i <∞. Then, the following inequality holds:

β(2)1 (G) ≥ n− 1 +

n∑i=1

(β

(2)1 (Gi)−

1

|Gi|

)−∞∑j=1

1

wj.

A proof of this result was given in [14]. It can be used in many cases already ifthe groups Gi are isomorphic to Z or Z/pZ, see for example [13].

Another result says that the set of marked groups with first `2-Betti numbergreater or equal so some constant is closed in Grigorchuk’s space of marked groups,see [15] for definitions and further references. More precisely, we have:

Theorem 6 (Pichot, see [15]). Let ((Gn, S))n∈N be a convergent sequence ofmarked groups in Grigorchuk’s space of marked groups. Then,

β(2)1 (G) ≥ lim sup

n→∞β

(2)1 (Gn).

This applies in particular to limits of free groups and shows that they all havea positive first `2-Betti number. In particular, there is an abundance of finitelypresented groups with positive first `2-Betti number.

4 Normal generation by torsion elementsThe first main result in this note extends the trivial upper bound from Equation(1) (under some additional hypothesis) to the case where the group is normallygenerated by a certain finite set of elements. The additional hypothesis is thatthe group G be finitely presented and residually p-finite for some prime p. Moreprecisely:

Theorem 7. Let G be an infinite, finitely presented, residually p-finite group forsome prime p. If G is normally generated by a subgroup Λ, then

β(2)1 (G) ≤ β(2)

1 (Λ).

6 Andreas Thom

In particular, if G is normally generated by elements g1, . . . , gk ∈ G of ordern1, . . . , nk ∈ 1, 2, . . . ∪ ∞, then

β(2)1 (G) ≤ k − 1−

k∑i=1

1

ni. (2)

Proof. Let gi | i ∈ N be a generating set for Λ. Let H be a finite index normalsubgroup of G, such that G/H is of p-power order. We consider Z1(G,Z[G/H]),the abelian group of 1-cocycles of the group G with values in the G-module Z[G/H].In a first step, we will show that the restriction map

σ : Z1(G,Z[G/H])→ Z1(Λ,Z[G/H])

is injective.Note that there is a natural injective evaluation map

π : Z1(Λ,Z[G/H])→ Z[G/H]⊕∞

which sends a 1-cocycle c to the values on the gi, i.e. c 7→ (c(gi))∞i=1.

We claim that π σ is injective. Indeed, assume that c ∈ ker(π σ) and assumethat c(g) ∈ ωm for all g ∈ G, where m is some integer greater than or equal zero.Since g is in the normal closure of gi | i ∈ N, there exists some natural numberl ∈ N and h1, . . . hl ∈ G, such that

g =

l∏i=1

hig±1q(i)h

−1i ,

for some function q : 1, . . . , l → N. Computing c(g) using the cocycle relationand c(g±i ) = 0, for 1 ≤ i ≤ k, we get

c(g) =

l∑i=1

(i−1∏j=1

hjg±q(j)h

−1j

)(1− hig±q(i)h

−1i

)c(hi).

By hypothesis c(hi) ∈ ωm for 1 ≤ i ≤ l and we conclude that c(g) ∈ ωm+1. Thisargument applies to all g ∈ G. Since the hypothesis c(g) ∈ ωm is obviously satisfiedfor m = 0, we finally get by induction that c(g) ∈ ωm for all m ∈ N and hence

c(g) ∈∞⋂m=1

ωm, ∀g ∈ G.

By Theorem 3 and since G/H is of prime power order, we know that⋂∞m=1 ω

m =0. Hence, c(g) = 0 for all g ∈ G. This proves that the map π σ and hence σ isinjective.

Note that

dimCH1(G,C[G/Hn]) = dimC Z

1(G,C[G/Hn])− [G : H] + 1


and also

dimCH1(Λ,C[G/Hn]) = dimC Z

1(Λ,C[G/Hn])− [G : H] + 1,

as Λ surjects onto G/H, using that a finite p-group cannot be normally generatedby a proper subgroup.

Now, by assumption, there exists a chain

· · · ⊂ Hn+1 ⊂ Hn ⊂ · · · ⊂ G

of finite index subgroups with p-power index such that

∞⋂n=1

Hn = e.

The claim is now implied by Lück’s Approximation Theorem (see Theorem 4).Indeed, Lück’s Approximation Theorem applied to the chain of finite index sub-groups of p-power index gives:

β(2)1 (G) = lim

n→∞

dimC H1(G,C[G/Hn])

[G : Hn]≤ limn→∞

dimC H1(Λ,C[G/Hn])

[G : Hn]≤ β(2)

1 (Λ).

Here, we used Kazhdan’s inequality in the last step (see [12, Theorem 1.1] fora proof). This finishes the proof of the first inequality.

The second claim follows from a simple and well-known estimate for the first`2-Betti number (see for example Theorem 3.2 in [14]) that we apply to Λ. Moredirectly, for each H, we can estimate the dimension of the image of π ⊗Z C. Sincegi has order ni, we compute

0 = c(gnii ) =

ni−1∑j=0

gji

c(gi),

where we denote by gi the image of gi in G/H. If the order of the image of gi ismi, then n−1

i

∑ni−1j=0 gji is a projection of normalized trace m−1

i such that

dimC (im(π)⊗Z C) ≤ [G : H] ·k∑i=1

(1− 1

mi

)≤ [G : H] ·

k∑i=1

(1− 1

ni

).

This implies that

dimC H1(Λ,Z[G/H])⊗Z C ≤ [G : H] ·

k∑i=1

(1− 1

ni

)− [G : H] + 1.

We now apply the first result. This finishes the proof, again using Lück’s Approx-imation Theorem.

8 Andreas Thom

Remark 2. Let G be an infinite, residually p-finite group. It follows from Proposi-tion 3.7 in [9] in combination with Lück’s Approximation Theorem that

β(2)1 (G) ≤ dimZ/pZH

1(G,Z/pZ)− 1.

This implies that β(2)1 (G) ≤ k− 1 in the situation that G is normally generated by

g1, . . . , gk. Our result improves this estimate in the case when some of the elementsg1, . . . , gk have finite order.

Remark 3. Consider G = PSL(2,Z) = 〈a, b | a2 = b3 = e〉. Then, β(2)1 (G) = 1

6 6= 0and G is normally generated by the element ab ∈ G. Hence, the assumption thatG is residually a p-group cannot be omitted in Theorem 7.

5 An uncertainty principle and applicationsIn this section we want to prove a quantitative estimate on the girth of a markedgroup in terms of its first `2-Betti number. In [13], Osin and the author constructedfor given ε > 0 a k-generated simple groups with first `2-Betti number greaterthan k− 1− ε. The construction involved methods from small cancellation theoryand in particular, those groups did not admit any short relations in terms of thenatural generating set. This in fact follows already from the main result in [15]. If

(Gi, Si)i∈N is a sequence of marked groups with |Si| = k and limi→∞

β(2)1 (Gi) = k−1,

then necessarilylimi→∞

girth(Gi, Si) =∞,

where girth(G,S) denotes the length of the shortest cycle in the Cayley graph of Gwith respect to the generating set S. Indeed, by [15], any limit point (G,S) of

the sequence (Gi, Si)i∈N satisfies β(2)1 (G) = k − 1 and hence is a free group on

the basis S. (This last fact is well known and is also a consequence of our nexttheorem.) In this section, we want to prove a quantitative version of this result.

Theorem 8. LetG be a finitely generated group with generating set S = g1, . . . , gk.Then,

girth(G,S) ≥ 1

k − 1− β(2)1 (G)

.

In order to prove this theorem, we need some variant of the so-called uncertaintyprinciple. We denote by ‖.‖ the usual operator norm on B(`2G) and use the samesymbol to denote the induced norm on CG, i.e., ‖f‖ = ‖λ(f)‖ for all f ∈ CG. The1-norm is denoted by ‖

∑g agg‖1 =

∑g|ag|. For f =

∑g agg we define its support

as supp := g ∈ G | ag 6= 0.

Theorem 9. Let G be a group and f ∈ CG be a non-zero element of the complexgroup ring. Then,

dimLG(f · LG) · |supp(f)| ≥ 1.


Proof. First of all we have dimLG(f · LG) = τ(pK), where K is the closure of theimage of λ(f) : `2G→ `2G.

τ(f∗f) ≤ dimLG(f · LG) · ‖f‖2 (3)

since

τ(f∗f) = τ(ff∗) = τ(pKff∗) ≤ τ(pK) · ‖ff∗‖ = dimLG(f · LG) · ‖f‖2.

Secondly, using the fact that ‖f‖22 = τ(f∗f), we see that

‖f‖21 ≤ |supp(f)| · τ(f∗f) (4)

by the Cauchy-Schwarz inequality applied to f ·χsupp(f), where the product here isthe pointwise product of coefficients and ‖f‖1 denotes the usual 1-norm on C[G].Combining Equations (3) and (4) we conclude

dimLG(f · LG) · |supp(f)| ≥(‖f‖1‖f‖

)2

. (5)

Now, since each group element acts as a unitary, and hence with operator norm1 on `2G, we get ‖f‖1 ≥ ‖f‖. This proves the claim and finishes the proof.

The preceding result and the following corollary were proved as result of aquestion by Efremenko on MathOverflow.

Corollary 1. LetG be a finite group and f ∈ C[G] be an arbitrary non-zero element.Then,

dimC(f · C[G]) · |supp(f)| ≥ |G|.

We are now ready to prove Theorem 8.

Proof. (Theorem 8) Again, we study the map π : Z1(G,LG) → LG⊕k, which isgiven by c 7→ (c(gi))

ki=1. If w ∈ Fk is some word such that w(g1, . . . , gk) = e in G,

then

0 = c(w(g1, . . . , gk)

)=

k∑i=1

∂i(w)(g1, . . . , gk) · c(gi),

where ∂i : Z[Fk] → Z[Fk] denotes the i-th Fox derivative for 1 ≤ i ≤ k. Thus, theimage of π lies is annihilated by the LG-linear map

(ξ1, . . . , ξk) 7→k∑i=1

∂i(w)(g1, . . . , gk)ξi .

In particular, the image of π does not intersect with K := ker(∂1(w)) ⊕ 0 ⊕· · · ⊕ 0. The number of summands in ∂1(w) is equal to the number of occurencesof the letters g±1 in w. Thus, we have and im(π) ∩ K = 0 and dimLG(K) ≥|supp(∂1(w))|−1. Thus

dimLG im(π) ≤ k − |supp(∂1(w))|−1 ≤ k − 1

`(w).

For any infinite group β(2)1 (G) = dimLG Z

1(G,LG)−1. This implies that β(2)1 (G) ≤

k − 1− 1/`(w) and finishes the proof.

10 Andreas Thom

AcknowledgmentsI want to thank Klim Efremenko, Mikhail Ershov, Ana Khukhro, and Denis Osinfor interesting comments. I thank the unknown referee for helpful comments. Thisresearch was supported by ERC-Starting Grant No.277728 Geometry and Analysisof Group Rings.

References

[1] M. Aschenbrenner, S. Friedl: 3-manifold groups are virtually residually p. Americanmathematical society (2013).

[2] M. F. Atiyah: Elliptic operators, discrete groups and von Neumann algebras. Astérisque32 (33) (1976) 43–72. Colloque “Analyse et Topologie” en l’Honneur de Henri Cartan(Orsay, 1974)

[3] G. Baumslag: Residually finite one-relator groups. Bull. Amer. Math. Soc. 73 (1967)618–620.

[4] J. Cheeger, M. Gromov: L2-cohomology and group cohomology. Topology 25 (2) (1986)189–215.

[5] E. Formanek: A short proof of a theorem of Jennings. Proc. Amer. Math. Soc. 26 (1970)405–407.

[6] K. Gruenberg: Residual properties of infinite soluble groups. Proc. London Math. Soc.(3) 7 (1957) 29–62.

[7] K. Gruenberg: The residual nilpotence of certain presentations of finite groups. Arch.Math. 13 (1962) 408–417.

[8] S. A. Jennings: The group ring of a class of infinite nilpotent groups. Canad. J. Math. 7(1955) 169–187.

[9] M. Lackenby: Covering spaces of 3-orbifolds. Duke Math. J. 136 (1) (2007) 181–203.

[10] W. Lück: Dimension theory of arbitrary modules over finite von Neumann algebras andL2-Betti numbers. II. Applications to Grothendieck groups, L2-Euler characteristics andBurnside groups. J. Reine Angew. Math. 496 (1998) 213–236.

[11] W. Lück: L2-invariants: theory and applications to geometry and K-theory.Springer-Verlag, Berlin (2002).

[12] W. Lück, D. Osin: Approximating the first L2-Betti number of residually finite groups.J. Topol. Anal. 3 (2) (2011) 153–160.

[13] D. Osin, A. Thom: Normal generation and `2-Betti numbers of groups. Math. Ann. 355(4) (2013) 1331–1347.

[14] J. Peterson, A. Thom: Group cocycles and the ring of affiliated operators. Invent. Math.185 (3) (2011) 561–592.

[15] M. Pichot: Semi-continuity of the first l2-Betti number on the space of finitely generatedgroups. Comment. Math. Helv. 81 (3) (2006) 643–652.

[16] V. P. Platonov: A certain problem for finitely generated groups. Dokl. Akad. NaukUSSR 12 (1968) 492–494.


Author’s address:Andreas Thom, Mathematisches Institut, U Leipzig, PF 100920, 04009 Leipzig,Germany

E-mail: andreas.thom a©math.uni-leipzig.de

Received: 19 August, 2014Accepted for publication: 1 January, 2015Communicated by: Stephen Glasby


A subclass of strongly clean rings

Orhan Gurgun, Sait Halicioglu and Burcu Ungor

Abstract. In this paper, we introduce a subclass of strongly clean rings. LetR be a ring with identity, J be the Jacobson radical of R, and let J# denotethe set of all elements of R which are nilpotent in R/J . An element a ∈ Ris called very J#-clean provided that there exists an idempotent e ∈ R suchthat ae = ea and a − e or a + e is an element of J#. A ring R is said tobe very J#-clean in case every element in R is very J#-clean. We provethat every very J#-clean ring is strongly π-rad clean and has stable rangeone. It is shown that for a commutative local ring R, A(x) ∈ M2

(R[[x]]

)is very J#-clean if and only if A(0) ∈ M2(R) is very J#-clean. Variousbasic characterizations and properties of these rings are proved. We obtaina partial answer to the open question whether strongly clean rings havestable range one.

This paper is dedicated to Professor Abdullah Harmanci on his 70th birthday

1 IntroductionThroughout this paper, all rings are associative with identity unless otherwisestated. Nicholson in [16] defined clean elements and clean rings, also in [17] Nichol-son and Zhou introduced strongly clean rings and Chen continued studying stronglyclean rings and introduced strongly J-clean rings in [5]. Other generalizations ofclean notion of rings are investigated by many authors ([4], [6], [10], [12]). Let Udenote the set of all invertible elements and J be the Jacobson radical of R. Inthis paper, the set of all elements of R which are nilpotent in R/J will be denotedby J#. Clearly, J ⊆ J#. Let a be an element of R. The element a is called cleanprovided that there exist e2 = e ∈ R and u ∈ U such that a = e+ u. The elementa is strongly clean if there exist e2 = e ∈ R and u ∈ U such that a = e + u andeu = ue. An element a is called very clean if there exists e2 = e ∈ R and u ∈ Usuch that a = e + u or a = −e + u and eu = ue. In general, a ∈ R is (strongly orvery) T -clean if and only if there exists an idempotent e ∈ R such that (ae = eaand) a− e (or potentially a+ e for very cleanness) is in the set related to T . Here,

2010 MSC: 15A13, 15B99, 16L99Key words: Very J#-clean matrix, very J#-clean ring, local ring.

14 Orhan Gurgun, Sait Halicioglu and Burcu Ungor

T ∈ Nil, J, J#, and the corresponding sets are Nil(R) (the set of all nilpotentelements of R), J and J#, respectively. A ring R is said to have stable range oneif given a, b ∈ R for which aR + bR = R, there exists y ∈ R such that a+ by ∈ U .One of the most important features of stable range one is the cancellation of re-lated modules from direct sums. We know that stable range one in endomorphismrings implies cancellation in direct sums, that is, if A, B, C are modules such thatA ⊕ B ∼= A ⊕ C, and End(A) has stable range one, then B ∼= C [11, Theorem 2].Further, if R is directly finite, i.e., any x, y ∈ R satisfying xy = 1 also satisfyyx = 1, then so is Mn(R) (for details one can see [6]). But so far it is unknownwhether strongly clean rings have stable range one (see [17]). This motivates us toconstruct a natural subclass of strongly clean rings, namely, very J#-clean rings,which have stable range one.

Clearly, every commutative or Artinian strongly nil clean ring is strongly J-clean. But the converse is not true in general (see [5] or Example 2). Since Nil(R) ⊆J# and J ⊆ J#, we know that strongly J-clean rings and strongly nil-clean ringsare strongly J#-clean, and every very nil-clean ring is very J#-clean. Example 2is a very J#-clean ring, which is not very nil-clean. Every strongly J#-clean ringis very J#-clean but Example 3 is very J#-clean, which is not strongly J#-clean.Any very J#-clean ring is strongly clean (see Theorem 1) but there exists a stronglyclean ring which is not very J#-clean (e.g. Z5). Every strongly clean ring is veryclean. Example 4 is a very clean ring, which is not strongly clean. Now we illustraterelations between these classes of rings in the following:

Strongly J-clean // Strongly J#-clean // Very J#-clean

Strongly nil-clean

55

// Very nil-clean

55

Strongly clean

Very clean

None of the implications in the diagram are reversible.The paper is organized as follows: in Section 2, basic properties of very J#-

clean rings are given. We give some examples concerning their relations with cleanrings, strongly clean rings, strongly J#-clean rings. Further, we prove that if R isvery J#-clean, then R has stable range one. In Section 3, we construct severalexamples of very J#-clean rings. For instance, if R is an abelian very J#-cleanring, then the ring R[[x]] of power series over R is very J#-clean. In Section 4,we characterize the very J#-cleanness of matrices over commutative local rings.Further, we consider very J#-clean power series rings over such matrix rings.

In what follows, for a positive integer n, Zn and N denote the ring of integersmodulo n and the natural numbers, while for a prime integer p, Z(p) denotes thering of integers localized at the prime ideal (p), and we write Mn(R) for the ringsof all n×n matrices over a ring R. We write R[[x]] and Nil(R) for the ring of powerseries over R and the set of all nilpotent elements of R, respectively. Let R denotethe quotient ring R/J .

A subclass of strongly clean rings 15

2 Elementary resultsRecall that a ring R is called local if it has only one maximal left ideal (equivalently,maximal right ideal). It is well known that a ring R is local if and only if a+ b = 1in R implies that either a or b is invertible if and only if R is a division ring. Aring R is said to be reduced if it has no non-zero nilpotent elements. Now we beginwith the simple result.

Lemma 1. For a ring R we have that R is reduced if and only if J# = J . Inparticular, J# = J if R is commutative or local or R is the direct sum of divisionrings.

It is clear from Lemma 1 that if R is a commutative or local ring, then a ∈ Ris strongly J#-clean if and only if a ∈ R is strongly J-clean. Recall that a ring Ris called uniquely clean if every element can be written uniquely as the sum of anidempotent and a unit (see [18]).

Lemma 2. Let R be a direct sum of division rings. Then the following are equiv-alent.

(1) R is strongly J#-clean.

(2) R is a direct sum of two-element fields.

Proof. Note that if R is a direct sum of division rings (every local ring or commu-tative Artinian ring has this property), then R is (strongly, very) J#-clean if andonly if R is (strongly, very, respectively) J-clean, because, by Lemma 1, we haveJ# = J . Let Fn denote the field with n elements.

(1) ⇒ (2) Since R is strongly J-clean, we have R is Boolean, and so R ∼= ⊕F2

because R is a direct sum of division rings.(2) ⇒ (1) Assume that R is a direct sum of two-element fields. Then R is

uniquely clean by [18, Corollary 16]. This implies that R is abelian (that is, allidempotents in R are central) and for all a ∈ R there exists a unique idempotente ∈ R such that e− a ∈ J by [18, Theorem 20]. Thus R is strongly J-clean.

One may suspect that if R is a direct sum of two- or three-element fields, thenR is very J#-clean. The following example shows that this is not true in general.

Example 1. Let R denote the ring Z9 ⊕ Z9. Then we have R = Z3 ⊕ Z3 and theonly idempotents of the ring R are (0, 0), (1, 0), (0, 1), (1, 1). Further, note thatJ# = J . Hence (2, 4) ∈ R is not (strongly) very J#-clean.

(Strongly) Nil-clean elements (rings) are introduced by Diesl in [9], [10]. Clearly,every strongly nil-clean element (ring) is a strongly J#-clean element (ring). Butthere exists a strongly J#-clean element (ring) which is not strongly nil-clean ele-ment (ring) as the following example shows (see [5]).

Example 2. Let R =∞∏n=1

Z2n . For each n ∈ N, Z2n is a local ring with the

maximal ideal 2Z2n . Then Z2n/2Z2n∼= Z2. Hence R is strongly J-clean, and so R


is strongly J#-clean (and very J#-clean). Since the element r = (0, 2, 2, . . .) ∈ Ris not strongly nil-clean (and not very nil-clean), R is not strongly nil-clean (andnot very nil-clean).

Every strongly J#-clean (strongly J-clean) ring is very J#-clean (very J-clean)but there exists a very J#-clean (very J-clean) ring which is not strongly J#-clean(strongly J-clean) as the following example shows.

Example 3. The ring Z3 is very J#-clean which is not strongly J#-clean.

Proof. Let R = Z3. Note that R is strongly (or very) J#-clean if and only if Ris strongly (or very) J-clean because R is commutative, and we have J = J# = 0by Lemma 1. Since R is not Boolean, R is not strongly J#-clean, but R is veryJ#-clean.

Very clean elements (rings) are introduced by Chen et al. in [8]. Thus any veryJ#-clean ring is very clean. But the converse need not be true in general as shownbelow.

Example 4. Z(3) ∩ Z(5) is a very clean ring which is not very J#-clean.

Proof. Set R = Z(3)∩Z(5). If R is very J#-clean, then, by Theorem 1, it is stronglyclean, but it is not strongly clean by [8, Theorem 3.5] or by [2, Example 17].

The next result shows that for an element of a ring, being very J#-clean andstrongly J#-clean coincide under some conditions.

Proposition 1. Let R be a ring, 2 ∈ J , and a ∈ R. Then a is very J#-clean if andonly if it is strongly J#-clean.

Proof. If a ∈ R is strongly J#-clean, then it is very J#-clean. Conversely, assumethat a ∈ R is very J#-clean. Then there exist an idempotent e ∈ R and v ∈ J#

such that ae = ea and a = e+v or a = −e+v. If a = −e+v, then a = e+(v−2e).As 2 ∈ J , it easy to verify that v − 2e ∈ J#, hence a ∈ R is strongly J#-clean.This completes the proof.

Remark 1. If u is invertible, v ∈ J# and uv = vu, then we have that u + v andu− v is invertible.

Proof. Since v ∈ J# if and only if −v ∈ J#, we only need to prove one of u+v ∈ Uand u−v ∈ U . We prove that u−v ∈ U . Now, we have vn ∈ J , thus 1−u−nvn ∈ U .Now

1− u−nvn = 1− (u−1v)n = (1− u−1v)(1 + u−1v + · · ·+ (u−1v)n−1

).

Hence 1− u−1v is invertible, and so u− v = u(1− u−1v) ∈ U , because u ∈ U .

By the following result, we determine the set of all invertible elements of a veryJ#-clean ring.


Proposition 2. If R is a very J#-clean ring, then

U = u ∈ R | u− 1 ∈ J# or u+ 1 ∈ J# .

Proof. Let u ∈ U . Since R is very J#-clean, there exist an idempotent e ∈ R andv ∈ J# such that ue = eu and u = e + v or u = −e + v. Assume that u = e + v.Then u− v = e ∈ U implies that e = 1 and so u = v+ 1. Assume that u = −e+ v.Then v − u = e ∈ U implies that e = 1 and so u = v − 1.

On the other hand, suppose that u = v − 1 where v ∈ J#. Then we can findsome n ∈ N such that vn ∈ J . Hence 1 − avn ∈ U for any a ∈ R. If 1 − vn ∈ U ,then 1− v ∈ U because

1− vn = (1− v)(1 + v + · · ·+ vn−1),

and so u ∈ U . Suppose that u = v+1 where v ∈ J#. Then we can find some n ∈ Nsuch that vn ∈ J . Therefore 1 − avn ∈ U for any a ∈ R. If 1 + (−1)n−1vn ∈ U ,then 1 + v ∈ U because

1 + (−1)n−1vn = (1 + v)(1− v + · · ·+ (−1)n−1vn−1),

and so u ∈ U . Hence

U = u ∈ R | u− 1 ∈ J# or u+ 1 ∈ J#,

as required.

Every very nil-clean ring is very J#-clean, but there exists a very J#-clean ringwhich is not very nil-clean (see Example 2). Clearly, if J is nil, then a ∈ R is veryJ#-clean if and only if a ∈ R is very nil-clean.

Now we give the relations among strongly cleanness, very nil-cleanness and veryJ#-cleanness for the rings.

Theorem 1. Let R be a ring. If R is very J#-clean, then R is strongly clean andR is very nil-clean. If R is strongly clean, R is very nil-clean and 2 ∈ J#, then R isvery J#-clean.

Proof. Suppose that R is very J#-clean, and let a ∈ R. Then there exist anidempotent e ∈ R and v ∈ J# such that ae = ea and a = e + v or a = −e + v.This implies that a = (1 − e) + (2e − 1 + v) or a = 1 − e + v − 1. As ev = veand (2e − 1)−1 = 2e − 1, we get 2e − 1 + v ∈ U or v − 1 ∈ U by Remark 1 andProposition 2. Hence a ∈ R is strongly clean because 1− e is an idempotent. ThusR is strongly clean. Further, a = e+ v or a = −e+ v where vn = 0 for some n ∈ N.Therefore R is very nil-clean.

Assume that R is strongly clean, R is very nil-clean, 2 ∈ J# and let a ∈ R.Then there exists an idempotent e ∈ R such that a = e + u and ea = ae whereu ∈ U . As R is very nil-clean, we can find an idempotent f ∈ R such that uf = f uand u = f+w or u = −f+w where w ∈ R is nilpotent. Further, f = u−w ∈ U

(R)


or f = w− u ∈ U(R), and then f = 1. Hence u = 1 +w+ r or u = −1 +w+ r for

some r ∈ J . Therefore

a = e+ u = e+ 1 + w + r = (1− e) + (2e+ w + r)

ora = e+ u = e− 1 + w + r = −(1− e) + (w + r).

Obviously, (w+ r)m ∈ J or (2e+w+ r)m ∈ J for some m ∈ N. Consequently, R isvery J#-clean.

Recall that an element a ∈ R is called strongly π-rad clean provided that thereexists an idempotent e ∈ R such that ae = ea and a− e ∈ U and (eae)n = eane ∈J(eRe) for some integer n ≥ 1. A ring R is said to be strongly π-rad clean in caseevery element in R is strongly π-rad clean (see [9]). For instance, if R is local, thenit is strongly π-rad clean. It is well known that eJe = J(eRe) for any e2 = e ∈ R(see [13, Theorem 1.3.3]).

Theorem 2. If a ring R is very J#-clean, then it is strongly π-rad clean.

Proof. Let R be a very J#-clean ring and a ∈ R. Then there exist an idempotente ∈ R and v ∈ J# such that ae = ea and a = e + v or a = −e + v. Assume thata = e+v where vn ∈ J for some n ∈ N. This implies that a = (1−e)+(2e−1+v).As ev = ve and (2e − 1)−1 = 2e − 1, it is easy to verify that 2e − 1 + v ∈ U byRemark 1. Hence a(1− e) = (1− e)a and a− (1− e) ∈ U and

[(1− e)a(1− e)]n = [(1− e)v(1− e)]n = (1− e)vn(1− e) ∈ (1− e)J(1− e)

for some n ∈ N. Assume that a = −e + v where vm ∈ J for some m ∈ N.This implies that a = (1 − e) + (v − 1). By Proposition 2, v − 1 ∈ U . Thusa(1− e) = (1− e)a and a− (1− e) ∈ U and

[(1− e)a(1− e)]m = [(1− e)v(1− e)]m = (1− e)vm(1− e) ∈ (1− e)J(1− e)

for some m ∈ N. Therefore R is strongly π-rad clean, as asserted.

The converse of Theorem 2 need not be true as the following example shows.

Example 5. Since Z5 is a local ring, it is strongly π-rad clean, but not very J#-clean. Because 2 ∈ Z5 is not very J#-clean as J#(Z5) = J(Z5) = 0.

It is an open question that whether strongly clean rings have stable range one(see [17, Question 1]). In the next result, we obtain that very J#-clean rings havethis property. So by Theorem 3, we can give a partial answer to the open question.We know from [19] that a ring R has stable range one if and only if R has stablerange one. Recall that an element a of a ring R is called strongly π-regular if thereexist a positive integer n and x ∈ R such that an = an+1x. A ring R is said to bestrongly π-regular if every element of R is strongly π-regular. Ara showed that ifR is strongly π-regular, then R has stable range one (see [3, Theorem 4]).


Theorem 3. Let R be a very J#-clean ring. Then R is strongly π-regular, henceR has stable range one.

Proof. Let R be a very J#-clean ring and a ∈ R. Then there exist an idempotente ∈ R and v ∈ J# such that ae = ea and a = e + v or a = −e + v. Assume thata = e+v where vn ∈ J for some n ∈ N. This implies that an(1−e) = vn(1−e) ∈ Jand a = (1 − e) + (2e − 1 + v). As ev = ve and (2e − 1)−1 = 2e − 1, we getu := 2e−1+v ∈ U by Remark 1. Hence an = ane = une and an+1 = an+1e = un+1ein R. This gives an = an+1(u)−1 = (u)−1an+1, that is, a ∈ R is strongly π-regular.Suppose that a = −e+v where vm ∈ J for some m ∈ N. Write a = (1−e)+(v−1).This implies that am(1− e) = vm(1− e) ∈ J and

ame = (ae)m =((v − 1)e

)m= (v − 1)me.

Since vm ∈ J , we have v − 1 ∈ U . Hence am = ame = v − 1me and

am+1 = am+1e = ¯v − 1n+1

e

in R. This givesam = am+1(v − 1)−1 = (v − 1)−1am+1,

that is, a ∈ R is strongly π-regular, and so R is strongly π-regular. Thus R hasstable range one from [3, Theorem 4]. By the remark above, R has stable rangeone.

Let R be a ring and a ∈ R. Set

annl(a) = r ∈ R | ra = 0

andannr(a) = r ∈ R | ar = 0.

Then we have the following lemma.

Lemma 3. Let R be a ring and a = e+v or a = −e+v very J#-clean decompositionof a in R. Then annl(a) ⊆ annl(e) and annr(a) ⊆ annr(e).

Proof. Let r ∈ annl(a). Then ra = 0. Since ev = ve, we have re = rv or re = −rv,and so re = rve = rev or re = −rve = −rev. It follows that re(1 − v) = 0 orre(1 + v) = 0, and so re = 0 because 1 + v, 1 − v ∈ U . That is, r ∈ annl(e).Therefore annl(a) ⊆ annl(e). Similarly, we can prove that annr(a) ⊆ annr(e).

Theorem 4. Let R be a ring and f ∈ R be an idempotent. Then a ∈ fRf is veryJ#-clean in R if and only if a is very J#-clean in fRf .

Proof. Suppose a = e + v, e2 = e ∈ fRf , v ∈ J#(fRf), and ev = ve. Obviously,v ∈ J# because vn ∈ J(fRf) = fJf ⊆ J for some n ∈ N. Hence a ∈ fRf isvery J#-clean in R. Similarly, one can show that if a = −e + v, e2 = e ∈ fRf ,v ∈ J#(fRf), and ev = ve, then a ∈ fRf is very J#-clean in R.


Conversely, suppose that a = −e + v, e2 = e ∈ R, v ∈ J#, and ev = ve. Asa ∈ fRf , we see that

1− f ∈ annl(a) ∩ annr(a) ⊆ annl(e) ∩ annr(e).

Hence (1− f)v = 0 = v(1− f) and fv = vf = v. We observe that a = fef + fvf ,(fef)2 = fef , and

(fvf)m = vm ∈ fJf = J(fRf) ⊆ J#(fRf)

for some m ∈ N. Furthermore,

(fef)(fvf) = fevf = fvef = (fvf)(fef).

Similarly, one can prove that a ∈ fRf is very J#-clean in fRf where a = e + v,e2 = e ∈ R, v ∈ J#, and ev = ve. Therefore the proof is completed.

Corollary 1. A ring R is very J#-clean if and only if eRe is very J#-clean for anyidempotent e ∈ R.

Proof. Let a ∈ eRe. Since R is very J#-clean, we see that a ∈ eRe is very J#-cleanin R. According to Theorem 4, a ∈ eRe is very J#-clean in eRe. The converse isclear by using e = 1.

As is well known, every homomorphic image of a (strongly) clean ring is(strongly) clean (see [12], [16], [17]). Analogously, we can give the following re-sult.

Proposition 3. Every homomorphic image of very J#-clean rings is very J#-clean.

Proof. Let R be a very J#-clean ring and ϕ : R → S a surjective ring homomor-phism. Then for any b ∈ S, there exists a ∈ R such that ϕ(a) = b. Since R is veryJ#-clean, we can find an idempotent e ∈ R and v ∈ J# such that ae = ea anda = e+v or a = −e+v. Assume that a = −e+v and vn ∈ J for some n ∈ N. Thenϕ(a) = −ϕ(e) + ϕ(v) and ϕ(a)ϕ(e) = ϕ(e)ϕ(a). Obviously, (ϕ(e))2 = ϕ(e) ∈ S.Since ϕ(J) ⊆ J(S), we have ϕ(vn) = ϕ(v)n ∈ J(S) and so ϕ(v) ∈ J#(S). Similarly,one can show that ϕ(a) = ϕ(e) + ϕ(v) ∈ S is very J#-clean in S where a = e + vand v ∈ J#.

If I is a left ideal of a ring R, idempotents lift modulo I if, given a ∈ R witha2 − a ∈ I, there exists e2 = e ∈ R such that a − e ∈ I (see [16]). Note that R isa clean ring if and only if R/J is a clean ring and idempotents lift modulo J (see[12, Proposition 6]). Recall that a ring R is called abelian if every idempotent iscentral.

Theorem 5. Let I be an ideal of an abelian ring R with I ⊆ J . Then R is veryJ#-clean if and only if R/I is very J#-clean and idempotents lift modulo I.


Proof. Assume that R is very J#-clean. Then R/I is very J#-clean by Propo-sition 3. Further, by Theorem 1, R is strongly clean, and so idempotents liftmodulo I by [12, Proposition 6].

Conversely, suppose that R/I is very J#-clean and idempotents lift modulo Iand let a ∈ R. By assumption, for a ∈ R/I, there exists an idempotent e ∈ R/Isuch that ae = ea and a − e or a + e is an element of J#(R/I). Assume thata = −e + v where v ∈ J#(R/I). Then we can find some t ∈ N such that vt ∈J(R/I) = J/I and so v ∈ J#. Since idempotents lift modulo I, we may assumethat e2 = e. Hence a+ e− v ∈ I ⊆ J and so a is a very J#-clean element becausee is central. Similarly, one can prove that if a = e+ v and v ∈ J#(R/I), then a isa very J#-clean element.

3 ExamplesThe purpose of this section is to construct several examples for very J#-clean rings.

Let R be a ring and σ be an endomorphism of R. Let R[[x, σ]] be the set of all

power series over the ring R. For any∞∑i=0

aixi,∞∑i=0

bixi ∈ R[[x, σ]], we define

∞∑i=0

aixi +

∞∑i=0

bixi =

∞∑i=0

(ai + bi)xi,

and ( ∞∑i=0

aixi)( ∞∑

i=0

bixi)

=

∞∑i=0

cixi

where ci =i∑

k=0

akσk(bi−k

). Then R[[x, σ]] is a ring under the preceding addition

and multiplication. Clearly, R[[x, σ]] is R[[x]] only when σ is the identity morphism.Furthermore, J

(R[[x, σ]]

)= J + xR[[x, σ]] (see [14, Ex. 5.6]).

Lemma 4. If R[[x, σ]] is abelian, then σ(e) = e for every idempotent e ∈ R.

Proof. Since R[[x, σ]] is abelian, we have xe = ex for every idempotent e ∈ R.Hence we get xe = ex = σ(e)x, and so σ(e) = e, as asserted.

Proposition 4. Let R[[x, σ]] be an abelian ring. Then the following are equivalent.

(1) R is very J#-clean.

(2) R[[x, σ]] is very J#-clean.

Proof. (1)⇒ (2) Let a(x) ∈ R[[x, σ]]. Then we can find an idempotent e ∈ R andv ∈ J# such that a(0) = e+ v or a(0) = −e+ v. Assume that a(0) = e+ v. Thena(x) = e + v(x) where v(x) = a(x) − e = v + a1x + a2x

2 + · · · . Since σ(e) = efor any idempotent e ∈ R by Lemma 4, we see that ev(x) = v(x)e. Further, weconclude that v(x) ∈ J#

(R[[x, σ]]

)because v ∈ J# and

J(R[[x, σ]]

)= J + xR[[x, σ]].


This implies that a(x) ∈ R[[x, σ]] is very J#-clean. Assume that a(0) = −e + v.Similarly, we can show that a(x) ∈ R[[x, σ]] is very J#-clean. Thus R[[x, σ]] is veryJ#-clean.

(2) ⇒ (1) Let a ∈ R. Then we can find an idempotent e(x) ∈ R[[x, σ]] andv(x) ∈ J#

(R[[x, σ]]

)such that ae(x) = e(x)a and a = e(x) + v(x) or a = −e(x) +

v(x). Obviously, e(0) ∈ R is an idempotent and v(0) ∈ J#. Since a = e(0) + v(0)or a = −e(0) + v(0) and ae(0) = e(0)a, we obtain that a ∈ R is very J#-clean, andtherefore R is very J#-clean.

Remark 2. As in the proof of [1, Lemma 2.18], we can show that the idempotentsof R[[x, σ]] belong to R. Hence if R is abelian, then so is R[[x, σ]].

The next result is a characterization of being very J#-clean for abelian rings.

Theorem 6. Let R be an abelian ring. Then the following conditions are equiva-lent.


(2) R[[x]]/〈xn〉 is very J#-clean for all n ≥ 2.

(3) R[[x]]/〈x2〉 is very J#-clean.

(4) R[x]/〈x2〉 is very J#-clean.

Proof. (1)⇒ (2) IfR is very J#-clean, thenR[[x]] is very J#-clean by Proposition 4and so R[[x]]/〈xn〉 is very J#-clean by Proposition 3 for all n ≥ 2.

(2)⇒ (3) It is clear.(3) ⇒ (1) Since R is abelian, so is R[[x]] by Remark 2. Note that J(R[[x]]) =

J + xR[[x]]. Then 〈x2〉 ⊆ J(R[[x]]), and so R is very J#-clean by Theorem 5.(3)⇔ (4) Since R[x]/〈x2〉 ∼= R[[x]]/〈x2〉, there is nothing to show.

Let R be a ring and σ : R→ R be an endomorphism. Set

D2(R, σ) =

(a b0 a

) ∣∣∣∣ a, b ∈ R ,addition and multiplication are defined as follows:(

a b0 a

)+

(c d0 c

)=

(a+ c b+ d

0 a+ c

);(

a b0 a

)(c d0 c

)=

(ac ad+ bσ(c)0 ac

).

Then D2(R, σ) is a ring with the identity

(1 00 1

). Denote D2(R, 1R) by D2(R),

where 1R : R→ R, r 7→ r. Further, it can be verified that

J(D2(R, σ)

)=

(a b0 a

) ∣∣∣∣ a ∈ J, b ∈ R .


Proposition 5. Let R be an abelian ring and σ : R → R be an endomorphism.Then the following are equivalent.


(2) D2(R, σ) is very J#-clean.

Proof. Note that since R is abelian, σ(e) = e for every idempotent e ∈ R byLemma 4 and Remark 2.

(1) ⇒ (2) Let A :=

(a b0 a

)∈ D2(R, σ). Then there exists an idempotent

e ∈ R such that ae = ea and v := a − e ∈ J# or v := a + e ∈ J#. Assume that

v := a−e ∈ J# and vn ∈ J for some n ∈ N. Since V n =

(vn ∗0 vn

)∈ J

(D2(R, σ)

)where V =

(v b0 v

),

A−(e 00 e

)= V ∈ J#

(D2(R, σ)

).

As R is abelian and σ(e) = e, we see that EA = AE where E2 = E =

(e 00 e

)(because EA = AE if and only if eb = bσ(e) = be). Therefore A ∈ D2(R, σ) is veryJ#-clean. Assume that v := a + e ∈ J#. Similar to the preceding discussion, itcan be shown that A ∈ D2(R, σ) is very J#-clean, as required.

(2) ⇒ (1) Let a ∈ R. Then A :=

(a 00 a

)∈ D2(R, σ). By hypothesis, there

exists an idempotent E :=

(e b0 e

)∈ D2(R, σ) such that AE = EA and

A+ E ∈ J#(D2(R, σ)

)or A − E ∈ J#

(D2(R, σ)

). As E is an idempotent, we have e = e2. Further, we

get ea = ae, and that a− e ∈ J# or a+ e ∈ J#. Therefore R is very J#-clean.

Let R be a ring and V an R-R-bimodule which is a general ring (possibly withno unity) in which (vw)r = v(wr), (vr)w = v(rw) and (rv)w = r(vw) hold forall v, w ∈ V and r ∈ R. Then ideal-extension (it is also called Dorroh extension)I(R;V ) of R by V is defined to be the additive abelian group I(R;V ) = R ⊕ Vwith multiplication (r, v)(s, w) = (rs, rw + vs+ vw).

Proposition 6. An ideal-extension S = I(R;V ) is very J#-clean if the followingconditions are satisfied.

(1) R is very J#-clean;

(2) If e2 = e ∈ R, then ev = ve for all v ∈ V ;

(3) If v ∈ V , then v + w + vw = 0 for some w ∈ V .

Furthermore, if S is very J#-clean, then R is very J#-clean.


Proof. Suppose that (1), (2) and (3) are satisfied. Let s = (r, w) ∈ S and (by (1))write r = e+v or r = −e+v, e2 = e, v ∈ J# and re = er. Assume that r = −e+vand vn ∈ J for some n ∈ N. Then s = −(e, 0) + (v, w) and (e, 0)2 = (e, 0) ∈ S.Note that (0, V ) ⊆ J(S) if and only if (3) holds (see [15]). Since (v, w)n = (vn, ∗),it suffices to show that (vn, 0) ∈ J(S). For any (p, q) ∈ S,

(1, 0)− (vn, 0)(p, q) = (1− vnp,−vnq) ∈ U(S)

because(1− vnp,−vnq) = (1− vnp, 0)(1, (1− vnp)−1(−vnq))

and(1, (1− vnp)−1(−vnq)) = (1, 0) + (0, (1− vnp)−1(−vnq)) ∈ U(S)

by (0, V ) ⊆ J(S). Thus (vn, 0) ∈ J(S) and so (v, w) ∈ J#(S). By (2), (r, w)(e, 0) =(e, 0)(r, w). The case where r = e+ v can be similarly handled.

On the other hand, suppose that S is very J#-clean and let a ∈ R. Then(a, 0) = (e, t) + (v, w) or (a, 0) = −(e, t) + (v, w), (e, t)2 = (e, t), (v, w) ∈ J#(S)and (a, 0)(e, t) = (e, t)(a, 0). Assume that (a, 0) = (e, t)+(v, w) and (v, w)m ∈ J(S)for some m ∈ N. Since (v, w)m ∈ J(S), (e, t)2 = (e, t) and (a, 0)(e, t) = (e, t)(a, 0),we get a = e+ v, vm ∈ J , e2 = e ∈ R, and ae = ea. Hence a is strongly J#-clean.Suppose (a, 0) = −(e, t) + (v, w) and (v, w)n ∈ J(S) for some n ∈ N. Similarly, itcan be shown that −a is strongly J#-clean and so R is very J#-clean.

Example 6. Let R be an abelian very J#-clean ring, n a positive integer and

S =

a a12 · · · a1n

0 a · · · a2n

......

. . ....

0 0 · · · a

∣∣∣∣∣∣∣∣∣∣a, aij ∈ R(i < j)

.

Then S is very J#-clean and noncommutative if n ≥ 3.

Proof. Let

V =

0 a12 · · · a1n

0 0 · · · a2n

......

. . ....

0 0 · · · 0

∣∣∣∣∣∣∣∣∣∣aij ∈ R(i < j)

.

Then S ∼= I(R;V ). By applying Proposition 6, (1) is clear; (2) holds because R isabelian and (3) follows because of V ⊆ J(S).

4 Very J#-clean 2 × 2 matricesLet f, g ∈ R[x] be polynomials over a commutative ring R and let (f, g) denotethe ideal generated by f, g. A polynomial f(x) ∈ R[x] is a monic polynomial ofdegree n if f(x) = xn + an−1x

n−1 + · · · + a1x + a0 where an−1, . . . , a1, a0 ∈ R. Ifϕ ∈ Mn(R), we use χ(ϕ) to stand for the characteristic polynomial det(xIn − ϕ).


The aim of this section is to characterize a single very J#-clean 2 × 2 matrixover a commutative local ring by means of the factorization of its characteristicpolynomial.

We begin with the following result from [7], and give the proof of it for the sakeof completeness.

Lemma 5. Let R be a commutative ring and ϕ ∈ Mn(R). Then the following areequivalent.

(1) ϕ ∈ J#(Mn(R)

).

(2) χ(ϕ) ≡ xn (mod J).

(3) There exists a monic polynomial h ∈ R[x] such that h ≡ xdeg h (mod J) forwhich h(ϕ) = 0.

Proof. Note that J(Mn(R)

)= Mn

(J)

and Mn(R)/J(Mn(R)

)= Mn(R). Further-

more, since R is commutative, we have that Nil(R) ⊆ J .(1) ⇒ (2) If ϕ ∈ J#

(Mn(R)

), then ϕ is nilpotent in Mn(R). According to

[4, Proposition 3.5.4], we get χ(ϕ) ≡ xn (mod Nil(R)). So χ(ϕ) ≡ xn (mod J)because Nil(R) ⊆ J .

(2) ⇒ (3) Set h = χ(ϕ). Then h ≡ xdeg h (mod J). By Cayley-HamiltonTheorem, h(ϕ) = 0.

(3) ⇒ (1) Assume that h = xn + an−1xn−1 + · · · + a1x + a0 where

ai ∈ J for 0 ≤ i ≤ n − 1. Then h ≡ xn (mod Nil(R)) and h(ϕ) = 0. Again,by [4, Proposition 3.5.4], ϕ is nilpotent in Mn(R). This gives ϕ ∈ J#

(Mn(R)

).

Definition 1. [7, Definition 2.4] For r ∈ R, define

Jr = f ∈ R[x] | f is monic, and f ≡ (x− r)deg f (mod J#).

Remark 3. If R is commutative, then J# is simply the Jacobson radical. So weget

Jr = f ∈ R[x] | f is monic, and f ≡ (x− r)deg f (mod J) .

By f ≡ (x − r)deg f (mod J), we mean f − (x − r)deg f ∈ J [x]. Furthermore, it iswell known that

χ(ϕ) = x2 − tr(ϕ)x+ det(ϕ) and χ(−ϕ) = x2 + tr(ϕ)x+ det(ϕ)

because tr(−ϕ) = − tr(ϕ) and det(ϕ) = det(−ϕ) for ϕ ∈ M2(R). In general, notethat

χ(−ϕ)(x) = det(xIn − (−ϕ)

)= (−1)n det

((−x)In + ϕ

)= (−1)nχ(ϕ)(−x)

and det(−ϕ) = (−1)n det(ϕ) for ϕ ∈Mn(R).

For an easy reference, we mention the following lemmas without proofs. Re-call that a commutative ring R is called projective-free if every finitely generatedprojective R-module is free. Any commutative local ring is projective-free.


Lemma 6. [7, Lemma 2.5] Let R be a projective-free ring and h ∈ R[x] a monicpolynomial of degree n, let ϕ ∈Mn(R). If h(ϕ) = 0 and there exists a factorizationh = h0h1 such that h0 ∈ J0 and h1 ∈ J1, then ϕ is strongly J#-clean.

Lemma 7. [7, Theorem 2.6] Let R be a projective-free ring and h ∈ R[x] a monicpolynomial of degree n. Then the following are equivalent.

(1) Every ϕ ∈Mn(R) with χ(ϕ) = h is strongly J#-clean.

(2) There exists a factorization h = h0h1 such that h0 ∈ J0 and h1 ∈ J1.

In the proof of Lemma 8 and Theorem 7, we refer to Lemma 6 and Lemma 7.

Lemma 8. Let R be a commutative local ring and h ∈ R[x] a monic polynomialof degree n, let ϕ ∈ Mn(R). If h(ϕ) = 0 and there exists a factorization h = h0h1

such that h0 ∈ J0 and h1 ∈ J1 ∪ J−1, then ϕ is very J#-clean.

Proof. By hypothesis, there exists a factorization h = h0h1 such that h0 ∈ J0 andh1 ∈ J1 ∪ J−1. If h1 ∈ J1, then ϕ is strongly J#-clean by Lemma 6, and so ϕ isvery J#-clean. Hence we assume that h0 ∈ J0 and h1 ∈ J−1. Then

h0 ≡ xdeg(h0) (mod J) and h1 ≡ (x− (−1))deg(h1) (mod J) .

Set t := −x and g(t) := (−1)deg(h)h(−t). Then g(t) factors as g = g0g1, where

g0(t) = (−1)deg(h0)h0(−t) and g1(t) = (−1)deg(h1)h1(−t) .

Note that deg(g0) = deg(h0) and deg(g1) = deg(h1). Since h0 ≡ xdeg(h0) (mod J),we see that

g0(t) = (−1)deg(h0)h0(−t) ≡ (−1)deg(h0)xdeg(h0) ≡ tdeg(g0) (mod J) ,

and so g0 ∈ J0. Further, as h1 ≡ (x− (−1))deg(h1) (mod J), we have

g1(t) = (−1)deg(h1)h1(−t) ≡ (−1)deg(h1)(−t− (−1))deg(h1)

≡ (t− 1)deg(g1) (mod J) ,

and so g1 ∈ J1. We observe that g(−ϕ) = 0 because h(ϕ) = 0. In view ofLemma 6, −ϕ ∈ Mn(R) is strongly J#-clean. That is, ϕ is very J#-clean. Theproof is completed.

Theorem 7. Let R be a commutative local ring and h ∈ R[x] a monic polynomialof degree n. Then the following are equivalent.

(1) Every ϕ ∈Mn(R) with χ(ϕ) = h is very J#-clean.

(2) There exists a factorization h = h0h1 such that h0 ∈ J0 and h1 ∈ J1 ∪ J−1.


Proof. (1) ⇒ (2) Since ϕ is very J#-clean, ϕ or −ϕ is strongly J#-clean. If ϕ isstrongly J#-clean, then there exists a factorization h = h0h1 such that h0 ∈ J0 andh1 ∈ J1 by Lemma 7. Suppose −ϕ is strongly J#-clean. It follows by Lemma 7that g(t) := χ(−ϕ) factors as g = g0g1 where g0 ∈ J0 and g1 ∈ J1. This implies

h(x) = χ(ϕ) = (−1)deg(h)g(−x) = (−1)deg(h)g0(−x)g1(−x) .

Set h0(x) = (−1)deg(g0)g0(−x) and h1(x) = (−1)deg(g1)g1(−x). Then h = h0h1.Since g0(t) ≡ tdeg(g0) (mod J), we get

h0(x) = (−1)deg(g0)g0(−x) ≡ xdeg(g0) (mod J) ,

hence h0 ∈ J0. In addition, as g1(t) ≡ (t− 1)deg(g1) (mod J), we see that

h1(x) = (−1)deg(g1)g1(−x) ≡ (x+ 1)deg(g1) (mod J) .

This gives h1 ∈ J−1. That is, h0 ∈ J0 and h1 ∈ J1 ∪ J−1, as asserted.(2) ⇒ (1) For any ϕ ∈ M2(R) with χ(ϕ) = h, we have h(ϕ) = 0 by the

Cayley-Hamilton Theorem. In light of Lemma 8, ϕ is very J#-clean.

Corollary 2. [7, Corollary 2.8] Let R be a commutative local ring and ϕ ∈M2(R).Then ϕ is strongly J#-clean if and only if

(1) χ(ϕ) ≡ x2 (mod J); or

(2) χ(ϕ) ≡ (x− 1)2 (mod J); or

(3) χ(ϕ) has a root in J and a root in 1 + J .

In analogy with Corollary 2, we have the following result.

Corollary 3. Let R be a commutative local ring and ϕ ∈ M2(R). Then −ϕ isstrongly J#-clean if and only if

(1) χ(ϕ) ≡ x2 (mod J); or

(2) χ(ϕ) ≡ (x+ 1)2 (mod J); or

(3) χ(ϕ) has a root in J and a root in −1 + J .

Proof. Suppose that −ϕ is strongly J#-clean. As in the proof of Theorem 7, thereexists a factorization χ(ϕ) = h0h1 such that h0 ∈ J0 and h1 ∈ J−1. Consider thefollowing cases:Case I. deg(h0) = 2 and deg(h1) = 0. Then h0 = χ(ϕ) = x2 − tr(ϕ)x+ det(ϕ)

and h1 = 1. As h0 ∈ J0, it follows from Lemma 5 that ϕ ∈ J#(M2(R)

)or

equivalently, χ(ϕ) ≡ x2 (mod J).Case II. deg(h0) = 0 and deg(h1) = 2. Then h1(x) = χ(ϕ) ≡ (x+1)2 (mod J)

because h1 ∈ J−1.Case III. deg(h0) = 1 and deg(h1) = 1. Then h0 = x − α and h1 = x − β.

Since h0 ∈ J0, we see that h0 ≡ x (mod J), and then α ∈ J . As h1 ∈ J−1, we have


h1 ≡ x + 1 (mod J), and so β ∈ −1 + J . Therefore χ(ϕ) has a root in J and aroot in −1 + J .

For the reverse implication, if (1) or (2) is valid, then −ϕ ∈ J#(M2(R)

)or

I2 + ϕ ∈ J#(M2(R)

). This implies that −ϕ is strongly J#-clean. Suppose that

χ(ϕ) has a root in J and a root in −1 + J and −ϕ, I2 + ϕ 6∈ J(M2(R)

). By

Remark 3, we know that χ(ϕ)(−x) = χ(−ϕ)(x). In this case, χ(ϕ) has a root in Jand a root in 1 + J . According to [6, Theorem 16.4.31], ϕ is strongly J-clean, andtherefore it is strongly J#-clean.

For instance, choose ϕ =

(0 78 1

)∈ M2(Z9). Note that J(Z9) = 3Z9. Then

χ(ϕ) = x2 + x+ 7 = (x+ 1)2 + 6x+ 6. Hence χ(ϕ) ≡ (x+ 1)2 (mod J(Z9)), andso ϕ ∈M2(Z9) is very J#-clean by Corollary 3.

In the next, we investigate very J#-clean matrices over power series rings.

Theorem 8. Let R be a commutative local ring. Then the following are equivalent.

(1) A(x) ∈M2

(R[[x]]

)is very J#-clean.

(2) A(0) ∈M2(R) is very J#-clean.

Proof. (1)⇒ (2) Since A(x) is very J#-clean in M2

(R[[x]]

), there exist an

E(x) = E2(x) ∈M2

(R[[x]]

)and V (x) ∈ J#

(M2(R[[x]])

)such that E(x)V (x) = V (x)E(x), and

A(x) = E(x) + V (x) or A(x) = −E(x) + V (x) .

This implies that E(0)V (0) = V (0)E(0) and

A(0) = E(0) + V (0) or A(0) = −E(0) + V (0) ,

where E(0) = E2(0) ∈ M2(R) and V (0) ∈ J#(M2(R)

). Hence A(0) is very J#-

clean in M2(R).(2)⇒ (1) Since R[[x]]/J

(R[[x]]

) ∼= R/J and R is local, R[[x]] is local. Assumethat −A(0) is strongly J#-clean. Then

• −A(0) ∈ J#(M2(R)

);

• or I2 +A(0) ∈ J#(M2(R)

);

• or the characteristic polynomial χ(A(0)

)= y2−µy+λ has a root α ∈ −1+J

and a root β ∈ J .

If −A(0) ∈ J#(M2(R)

), then

−A(x) ∈ J#(M2(R[[x]])

).


If I2 +A(0) ∈ J#(M2(R)

), then

I2 +A(x) ∈ J#(M2(R[[x]])

).

Otherwise, we write y =∞∑i=0

bixi and

χ(−A(x)) = y2 − µ(x)y + λ(x) .

Then y2 =∞∑i=0

cixi where ci =

i∑k=0

bkbi−k. Let

µ(x) =

∞∑i=0

µixi, λ(x) =

∞∑i=0

λixi ∈ R[[x]]

where µ0 = µ and λ0 = λ. Then

y2 − µ(x)y + λ(x) = 0

holds in R[[x]] if the following equations are satisfied:

b20 − b0µ0 + λ0 = 0;

(b0b1 + b1b0)− (b0µ1 + b1µ0) + λ1 = 0;

(b0b2 + b21 + b2b0)− (b0µ2 + b1µ1 + b2µ0) + λ2 = 0;

...

Obviously, µ0 = α + β ∈ U and α − β ∈ U . Let b0 = α. Since R is commutativeand 2b0 − µ0 = 2α− µ = α− β, there exists some b1 ∈ R such that

b1(2b0 − µ0) = b0µ1 − λ1.

Further, there exists some b2 ∈ R such that

b2(2b0 − µ0) = b0µ2 + b1µ1 − b21 − λ2.

By iteration of this process, we get b3, b4, . . . . Then y2 − µ(x)y + λ(x) = 0 hasa root y0(x) ∈ −1 + J

(R[[x]]

). If b0 = β ∈ J , analogously, we can show that

y2 − µ(x)y+ λ(x) = 0 has a root y1(x) ∈ J(R[[x]]

). In light of Corollary 3, −A(x)

is strongly J#-clean. Similarly, we can prove that if A(0) is strongly J#-clean,then A(x) is strongly J#-clean by Corollary 2. Therefore A(x) is very J#-clean inM2

(R[[x]]

).

Example 7. Let R = Z9[[x]] and

A(x) =

0 2−∞∑n=1

(1 + 5n)xn

1 1−∞∑n=1

(1 + 7n)xn

∈M2(R).


Then

A(0) =

(0 21 1

)= −

(1 00 1

)+

(1 21 2

),

where

(1 00 1

)is an idempotent and

(1 21 2

)∈ J#

(M2(Z9)

)because (

1 21 2

)2

=

(3 63 6

)∈ J

(M2(Z9)

).

Thus A(x) is very J#-clean by Theorem 8. Note that A(0) is not strongly J-clean.

Corollary 4. Let R be a commutative local ring and A(x) ∈ M2

(R[[x]]/(xm)

)(m ≥ 1). Then the following are equivalent.

(1) A(x) ∈M2

(R[[x]]/(xm)

)is very J#-clean.

(2) A(0) ∈M2(R) is very J#-clean.

Proof. (1)⇒ (2) is obvious.(2) ⇒ (1) Let ψ : R[[x]] → R[[x]]/(xm) denote the natural homomorphism.

Then ψ induces the surjective ring homomorphism

ψ∗ : M2

(R[[x]]

)→M2

(R[[x]]/(xm)

).

Then there exists B(x) ∈M2

(R[[x]]

)such that ψ∗

(B(x)

)= A(x). Then Theorem 8

completes the proof.

AcknowledgementThe authors would like to thank the referee(s) for his/her valuable suggestionswhich contributed to improve the presentation of this paper. The first authorthanks the Scientific and Technological Research Council of Turkey (TÜBITAK)for grant support.

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Authors’ address:Department of Mathematics, Ankara University, Turkey

E-mail: orhangurgun a©gmail.com, halici a©ankara.edu.tr, bungor a©science.ankara.edu.tr

Received: 12th September, 2013Accepted for publication: 13th April, 2015Communicated by: Attila Bérczes


Existence of solutions for Navier problems withdegenerate nonlinear elliptic equations

Albo Carlos Cavalheiro

Abstract. In this paper we are interested in the existence and uniquenessof solutions for the Navier problem associated to the degenerate nonlinearelliptic equations

∆(v(x) |∆u|q−2∆u)−n∑

j=1

Dj

[ω(x)Aj(x, u,∇u)

]= f0(x)−

n∑j=1

Djfj(x), in Ω

in the setting of the weighted Sobolev spaces.

1 IntroductionIn this paper we prove the existence and uniqueness of (weak) solutions in theweighted Sobolev space X = W 2,q(Ω, v) ∩ W 1,p

0 (Ω, ω) (see Definition 4) for theNavier problem

Lu(x) = f0(x)−∑n

j=1Djfj(x), in Ω

u(x) = 0, on ∂Ω

∆u(x) = 0, on ∂Ω

(1)

where L is the partial differential operator

Lu(x) = ∆(v(x) |∆u|q−2∆u)−

n∑j=1

Dj

[ω(x)Aj(x, u(x),∇u(x))

]where Dj = ∂/∂xj , Ω is a bounded open set in Rn, ω and v are two weightfunctions, ∆ is the usual Laplacian operator, 1 < p, q < ∞ and the functionsAj : Ω× R× Rn → R (j = 1, . . . , n) satisfy the following conditions:

2010 MSC: 35J70, 35J60Key words: degenerate nolinear elliptic equations, weighted Sobolev spaces, Navier problem

34 Albo Carlos Cavalheiro

(H1) x 7→ Aj(x, η, ξ) is measurable on Ω for all (η, ξ) ∈ R× Rn

(η, ξ) 7→ Aj(x, η, ξ) is continuous on R× Rn for almost all x ∈ Ω.

(H2) There exists a constant θ1 > 0 such that

[A(x, η, ξ)−A(x, η′, ξ′)] · (ξ − ξ′) ≥ θ1|ξ − ξ′|p

whenever ξ, ξ′ ∈ Rn, ξ 6= ξ′, where A(x, η, ξ) = (A1(x, η, ξ), . . . ,An(x, η, ξ))

(where a dot denote here the Euclidean scalar product in Rn).

(H3) A(x, η, ξ) · ξ ≥ λ1|ξ|p, where λ1 is a positive constant.

(H4) |A(x, η, ξ)| ≤ K1(x) + h1(x)|η|p/p′ + h2(x)|ξ|p/p′ , where K1, h1 and h2 arenon-negative functions, with h1 and h2 ∈ L∞(Ω), and K1 ∈ Lp

′(Ω, ω) (with

1/p+ 1/p′ = 1).

By a weight, we shall mean a locally integrable function ω on Rn such thatω(x) > 0 for a.e. x ∈ Rn. Every weight ω gives rise to a measure on the measurablesubsets on Rn through integration. This measure will be denoted by µ. Thus,µ(E) =

∫Eω(x) dx for measurable sets E ⊂ Rn.

In general, the Sobolev spaces Wk,p(Ω) without weights occur as spaces ofsolutions for elliptic and parabolic partial differential equations. For degeneratepartial differential equations, i.e., equations with various types of singularities inthe coefficients, it is natural to look for solutions in weighted Sobolev spaces (see[1], [2] and [4]).

In various applications, we can meet boundary value problems for elliptic equa-tions whose ellipticity is disturbed in the sense that some degeneration or sin-gularity appears. This bad behaviour can be caused by the coefficients of thecorresponding differential operator as well as by the solution itself. The so-calledp-Laplacian is a prototype of such an operator and its character can be interpretedas a degeneration or as a singularity of the classical (linear) Laplace operator (withp = 2). There are several very concrete problems from practice which lead tosuch differential equations, e.g. from glaceology, non-Newtonian fluid mechanics,flows through porous media, differential geometry, celestial mechanics, climatology,petroleum extraction, reaction-diffusion problems, etc.

A class of weights, which is particulary well understood, is the class of Ap-weights (or Muckenhoupt class) that was introduced by B. Muckenhoupt (see [11]).These classes have found many useful applications in harmonic analysis (see [13]).Another reason for studying Ap-weights is the fact that powers of distance to sub-manifolds of Rn often belong to Ap (see [10]). There are, in fact, many interestingexamples of weights (see [9] for p-admissible weights).

In the non-degenerate case (i.e. with v(x) ≡ 1), for all f ∈ Lp(Ω), the Poissonequation associated with the Dirichlet problem

−∆u = f(x), in Ω

u(x) = 0, on ∂Ω

Existence of solutions for Navier problems with degenerate nonlinear elliptic equations 35

is uniquely solvable in W 2,p(Ω) ∩ W 1,p0 (Ω) (see [8]), and the nonlinear Dirichlet

problem −∆pu = f(x), in Ω

u(x) = 0, on ∂Ω

is uniquely solvable in W 1,p0 (Ω) (see [3]), where ∆pu = div(|∇u|p−2∇u) is the

p- Laplacian operator. In the degenerate case, the weighted p-Biharmonic operatorhas been studied by many authors (see [12] and the references therein), and thedegenerated p-Laplacian was studied in [4].

The following theorem will be proved in section 3.

Theorem 1. Assume (H1) – (H4). If

(i) v ∈ Aq, ω ∈ Ap (with 1 < p, q <∞),

(ii) fj/ω ∈ Lp′(Ω, ω) (j = 0, 1, . . . , n),

then the problem (1) has a unique solution u ∈ X = W 2,q(Ω, v) ∩W 1,p0 (Ω, ω).

2 Definitions and basic resultsLet ω be a locally integrable nonnegative function in Rn and assume that0 < ω(x) < ∞ almost everywhere. We say that ω belongs to the Muckenhouptclass Ap, 1 < p < ∞, or that ω is an Ap-weight, if there is a positive constantC = Cp,ω such that(

1

|B|

∫B

ω(x) dx

)(1

|B|

∫B

ω1/(1−p)(x) dx

)p−1

≤ C

for all balls B ⊂ Rn, where |·| denotes the n-dimensional Lebesgue measure inRn. If 1 < q ≤ p, then Aq ⊂ Ap (see [7], [9] or [13] for more information aboutAp-weights). The weight ω satisfies the doubling condition if there exists a positiveconstant C such that µ(B(x; 2 r)) ≤ Cµ(B(x; r)), for every ball B = B(x; r) ⊂ Rn,where µ(B) =

∫Bω(x) dx. If ω ∈ Ap, then µ is doubling (see Corollary 15.7 in [9]).

As an example of Ap-weight, the function ω(x) = |x|α, x ∈ Rn, is in Ap if andonly if −n < α < n(p− 1) (see Corollary 4.4, Chapter IX in [13]).

If ω ∈ Ap, then (|E||B|

)p≤ C µ(E)

µ(B)

whenever B is a ball in Rn and E is a measurable subset of B (see 15.5 strongdoubling property in [9]). Therefore, if µ(E) = 0 then |E| = 0.

Definition 1. Let ω be a weight, and let Ω⊂Rn be open. For 0 < p <∞ we defineLp(Ω, ω) as the set of measurable functions f on Ω such that

‖f‖Lp(Ω,ω) =

(∫Ω

|f(x)|pω(x) dx

)1/p

<∞ .


If ω ∈ Ap, 1 < p < ∞, then ω−1/(p−1) is locally integrable and we haveLp(Ω, ω)⊂L1

loc(Ω) for every open set Ω (see Remark 1.2.4 in [14]). It thus makessense to talk about weak derivatives of functions in Lp(Ω, ω).

Definition 2. Let Ω ⊂ Rn be open, 1 < p < ∞ and ω ∈ Ap. We define theweighted Sobolev space W k,p(Ω, ω) as the set of functions u ∈ Lp(Ω, ω) with weakderivatives Dαu ∈ Lp(Ω, ω), 1 ≤ |α| ≤ k. The norm of u in W k,p(Ω, ω) is definedby

‖u‖Wk,p(Ω,ω) =

(∫Ω

|u(x)|pω(x) dx+∑

1≤|α|≤k

∫Ω

|Dαu(x)|pω(x) dx

)1/p

. (2)

We also define W k,p0 (Ω, ω) as the closure of C∞0 (Ω) with respect to the norm

‖·‖Wk,p(Ω,ω).

If ω ∈ Ap, then W k,p(Ω, ω) is the closure of C∞(Ω) with respect to the norm(2.1) (see Theorem 2.1.4 in [14]). The spaces W k,p(Ω, ω) and W k,p

0 (Ω, ω) are Ba-nach spaces and the spaces W k,2(Ω, ω) and W k,2

0 (Ω, ω) are Hilbert spaces.It is evident that a weight function ω which satisfies 0 < c1 ≤ ω(x) ≤ c2 for

x ∈ Ω (where c1 and c2 are constants), gives nothing new (the space Wk,p0 (Ω, ω) is

then identical with the classical Sobolev space Wk,p0 (Ω)). Consequently, we shall

be interested above in all such weight functions ω which either vanish somewherein Ω ∪ ∂Ω or increase to infinity (or both).

In this paper we use the following results.

Theorem 2. Let ω ∈ Ap, 1 < p < ∞, and let Ω be a bounded open set in Rn.If um→u in Lp(Ω, ω) then there exist a subsequence umk and a function Φ ∈Lp(Ω, ω) such that

(i) umk(x)→ u(x), mk →∞, µ-a.e. on Ω;

(ii) |umk(x)| ≤ Φ(x), µ-a.e. on Ω; (where µ(E) =∫Eω(x) dx).

Proof. The proof of this theorem follows the lines of Theorem 2.8.1 in [6].

Theorem 3. (The weighted Sobolev inequality) Let Ω be an open bounded set inRn and ω ∈ Ap (1 < p < ∞). There exist constants CΩ and δ positive such thatfor all u ∈ C∞0 (Ω) and all k satisfying 1 ≤ k ≤ n/(n− 1) + δ,

‖u‖Lkp(Ω,ω) ≤ CΩ‖∇u‖Lp(Ω,ω).

Proof. See Theorem 1.3 in [5].

Lemma 1. Let 1 < p <∞.

(a) There exists a constant αp such that∣∣∣|x|p−2x− |y|p−2y∣∣∣ ≤ αp|x− y|(|x|+ |y|)p−2

y ,

for all x, y ∈ Rn;


(b) There exist two positive constants βp, γp such that for every x, y ∈ Rn

βp(|x|+|y|)p−2|x−y|2 ≤(|x|p−2

x−|y|p−2y)·(x−y) ≤ γp

(|x|+|y|

)p−2|x− y|2.

Proof. See [3], Proposition 17.2 and Proposition 17.3.

Definition 3. Let Ω ⊂ Rn be a bounded open set and v ∈ Aq, ω ∈ Ap, 1 < p, q <

∞. We denote by X = W 2,q(Ω, v) ∩W 1,p0 (Ω, ω) with the norm

‖u‖X = ‖∇u‖Lp(Ω,ω) + ‖∆u‖Lq(Ω v).

Definition 4. We say that an element u ∈ X = W 2,q(Ω, v)∩W 1,p0 (Ω, ω) is a (weak)

solution of problem (1) if for all ϕ ∈ X we have∫Ω

|∆u|q−2∆u∆ϕv dx+

n∑j=1

∫Ω

ωAj(x, u(x),∇u(x))Djϕdx

=

∫Ω

f0 ϕdx+

n∑j=1

∫Ω

fj Djϕdx.

3 Proof of Theorem 1The basic idea is to reduce the problem (1) to an operator equation Au = T andapply the theorem below.

Theorem 4. Let A : X→X∗ be a monotone, coercive and hemicontinuous operatoron the real, separable, reflexive Banach space X. Then for each T ∈ X∗ theequation Au = T has a solution u ∈ X.

Proof. See Theorem 26.A in [16].

To prove the existence of solutions, we define B,B1, B2 : X × X → R andT : X → R by

B(u, ϕ) = B1(u, ϕ) +B2(u, ϕ) ,

B1(u, ϕ) =

n∑j=1

∫Ω

ωAj(x, u,∇u)Djϕdx =

∫Ω

ωA(x, u,∇u) · ∇ϕdx ,

B2(u, ϕ) =

∫Ω

|∆u|q−2∆u∆ϕv dx ,

T (ϕ) =

∫Ω

f0 ϕdx+

n∑j=1

∫Ω

fj Djϕdx .

Then u ∈ X is a (weak) solution to problem (1) if

B(u, ϕ) = B1(u, ϕ) +B2(u, ϕ) = T (ϕ), for all ϕ ∈ X.

Step 1. For j = 1, . . . , n we define the operator Fj : X → Lp′(Ω, ω) by

(Fju)(x) = Aj(x, u(x),∇u(x)).

We now show that operator Fj is bounded and continuous.


(i) Using (H4) we obtain

‖Fju‖p′

Lp′ (Ω,ω)=

∫Ω

|Fju(x)|p′ω dx =

∫Ω

|Aj(x, u,∇u)|p′ω dx

≤∫

Ω

(K1 + h1|u|p/p

′+ h2|∇u|p/p

′)p′

ω dx

≤ Cp∫

Ω

[(Kp′

1 + hp′

1 |u|p

+ hp′

2 |∇u|p)ω

]dx

= Cp

[ ∫Ω

Kp′

1 ω dx+

∫Ω

hp′

1 |u|pω dx+

∫Ω

hp′

2 |∇u|pω dx

], (3)

where the constant Cp depends only on p. We have, by Theorem 3,

∫Ω

hp′

1 |u|pω dx ≤ ‖h1‖p

′

L∞(Ω)

∫Ω

|u|p ω dx

≤ CpΩ‖h1‖p′

L∞(Ω)

∫Ω

|∇u|p ω dx

≤ CpΩ‖h1‖p′

L∞(Ω)‖u‖pX ,

and ∫Ω

hp′

2 |∇u|pω dx ≤ ‖h2‖p

′

L∞(Ω)

∫Ω

|∇u|pω dx ≤ ‖h2‖p′

L∞(Ω)‖u‖pX .

Therefore, in (3) we obtain

‖Fju‖Lp′ (Ω,ω) ≤ Cp(‖K‖Lp′ (Ω,ω) + (C

p/p′

Ω ‖h1‖L∞(Ω) + ‖h2‖L∞(Ω))‖u‖p/p′

X

),

and hence the boundedness.

(ii) Let um → u in X as m → ∞. We need to show that Fjum → Fju inLp′(Ω, ω). We will apply the Lebesgue Dominated Convergence Theorem. If

um → u in X, then um → u in Lp(Ω, ω) and |∇um| → |∇u| in Lp(Ω, ω).Using Theorem 2, there exist a subsequence umk and functions Φ1 and Φ2

in Lp(Ω, ω) such that

umk(x)→ u(x), µ1- a.e. in Ω,

|umk(x)| ≤ Φ1(x), µ1 - a.e. in Ω,

|∇umk(x)| → |∇u(x)|, µ1 - a.e. in Ω,

|∇umk(x)| ≤ Φ2(x), µ1 - a.e. in Ω.


where µ1(E) =∫Eω(x) dx. Hence, using (H4), we obtain

‖Fjumk − Fju‖p′

Lp′ (Ω,ω)=

∫Ω

|Fjumk(x)− Fju(x)|p′ω dx

=

∫Ω

|Aj(x, umk ,∇umk)−Aj(x, u,∇u)|p′ω dx

≤ Cp∫

Ω

(|Aj(x, umk ,∇umk)|p

′+ |Aj(x, u,∇u)|p

′)ω dx

≤ Cp[∫

Ω

(K1 + h1|umk |

p/p′+ h2|∇umk |p/p

′)p′

ω dx

+

∫Ω

(K1 + h1|u|p/p

′+ h2|∇u|p/p

′)p′

ω dx

]≤ 2Cp

∫Ω

(K1 + h1Φ

p/p′

1 + h2Φp/p′

2

)p′ω dx

≤ 2Cp

[∫Ω

Kp′

1 ω dx+

∫Ω

hp′

1 Φp1ω dx+

∫Ω

hp′

2 Φp2ω dx

]≤ 2Cp

[‖K1‖p

′

Lp′ (Ω,ω)+ ‖h1‖p

′

L∞(Ω)

∫Ω

Φp1ω dx

+ ‖h2‖p′

L∞(Ω)

∫Ω

Φp2ω dx

]≤ 2Cp

[‖K1‖p

′

Lp′ (Ω,ω)+ ‖h1‖p

′

L∞(Ω)‖Φ1‖pLp(Ω,ω)

+ ‖h2‖p′

L∞(Ω)‖Φ2‖pLp(Ω,ω)

].

By condition (H1), we have

Fjum(x) = Aj(x, um(x),∇um(x))→ Aj(x, u(x),∇u(x)) = Fju(x),

as m→ +∞. Therefore, by the Dominated Convergence Theorem, we obtain‖Fjumk − Fju‖Lp′ (Ω,ω) → 0, that is, Fjumk → Fju in Lp

′(Ω, ω). By the

Convergence principle in Banach spaces (see Proposition 10.13 in [15]) wehave

Fjum → Fju in Lp′(Ω, ω). (4)

Step 2. We define the operator G : X → Lq′(Ω, v) by

(Gu)(x) = |∆u(x)|q−2∆u(x).

We also have that the operator G is continuous and bounded. In fact,


(i) We have

‖Gu‖q′

Lq′ (Ω,v)=

∫Ω

∣∣|∆u|q−2∆u∣∣q′v dx

=

∫Ω

|∆u|(q−2)q′ |∆u|q′v dx

=

∫Ω

|∆u|qv dx ≤ ‖u‖qX .

Hence, ‖Gu‖Lq′ (Ω,v) ≤ ‖u‖q/q′

X .

(ii) If um → u in X then ∆um → ∆u in Lq(Ω, v). By Theorem 2, there exist asubsequence umk and a function Φ3 ∈ Lq(Ω, v) such that

∆umk(x)→ ∆u(x), µ2 - a.e. in Ω

|∆umk(x)| ≤ Φ3(x), µ2 - a.e. in Ω.

where µ2(E) =∫Ev(x) dx. Hence, using Lemma 1(a), we obtain, if q 6= 2

‖Gumk −Gu‖q′

Lq′ (Ω,v)=

∫Ω

|Gumk −Gu|q′v dx

=

∫Ω

∣∣∣∣|∆umk |q−2∆umk − |∆u|

q−2∆u

∣∣∣∣q′v dx

≤∫

Ω

[αq|∆umk −∆u|(|∆umk |+ |∆u|)(q−2)

]q′v dx

≤ αq′

q

∫Ω

|∆umk −∆u|q′(2Φ3)(q−2)q′v dx

≤ αq′

q 2(q−2)q′(∫

Ω

|∆umk −∆u|qv dx

)q′/q×(∫

Ω

Φ(q−2)qq′/(q−q′)3 v dx

)(q−q′)/q

≤ αq′

q 2(q−2)q′‖umk − u‖q′

X‖Φ‖q−q′Lq(Ω,v),

since (q − 2)qq′/(q − q′) = q if q 6= 2. If q = 2, we have

‖Gumk −Gu‖2L2(Ω,v) =

∫Ω

|∆umk −∆u|2v dx ≤ ‖umk − u‖2X .

Therefore (for 1 < q < ∞), by the Dominated Convergence Theorem, weobtain

‖Gumk −Gu‖Lq′ (Ω,v) → 0 ,

that is, Gumk → Gu in Lq′(Ω, v). By the Convergence principle in Banach

spaces (see Proposition 10.13 in [15]), we have

Gum → Gu in Lq′(Ω, v). (5)


Step 3. We have, by Theorem 3,

|T (ϕ)| ≤∫

Ω

|f0||ϕ|dx+

n∑j=1

∫Ω

|fj ||Djϕ|dx

=

∫Ω

|f0|ω|ϕ|ω dx+

n∑j=1

∫Ω

|fj |ω|Djϕ|ω dx

≤ ‖f0/ω‖Lp′ (Ω,ω)‖ϕ‖Lp(Ω,ω) +

n∑j=1

‖fj/ω‖Lp′ (Ω,ω)‖Djϕ‖Lp(Ω,ω)

≤(CΩ ‖f0/ω‖Lp′ (Ω,ω) +

n∑j=1

‖fj/ω‖Lp′ (Ω,ω)

)‖ϕ‖X .

Moreover, using (H4) and the Hölder inequality, we also have

|B(u, ϕ)| ≤ |B1(u, ϕ)|+ |B2(u, ϕ)|

≤n∑j=1

∫Ω

|Aj(x, u,∇u)||Djϕ|ω dx+

∫Ω

|∆u|q−2|∆u||∆ϕ|v dx . (6)

In (6) we have∫Ω

|A(x, u,∇u)| |∇ϕ|ω dx ≤∫

Ω

(K1 + h1|u|p/p

′+ h2|∇u|p/p

′)|∇ϕ|ω dx

≤ ‖K1‖Lp′ (Ω,ω)‖∇ϕ‖Lp(Ω,ω) + ‖h1‖L∞(Ω)‖u‖p/p′

Lp(Ω,ω)‖∇ϕ‖Lp(Ω,ω)

+ ‖h2‖L∞(Ω)‖∇u‖p/p′

Lp(Ω,ω)‖∇ϕ‖Lp(Ω,ω)

≤(‖K1‖Lp′ (Ω,ω) + (C

p/p′

Ω ‖h1‖L∞(Ω) + ‖h2‖L∞(Ω))‖u‖p/p′

X

)‖ϕ‖X ,

and ∫Ω

|∆u|q−2|∆u||∆ϕ|v dx =

∫Ω

|∆u|q−1|∆ϕ|v dx

≤(∫

Ω

|∆u|qv dx)1/q′(∫

Ω

|∆ϕ|qv dx)1/q

≤ ‖u‖q/q′

X ‖ϕ‖X .

Hence, in (6) we obtain, for all u, ϕ ∈ X

|B(u, ϕ)| ≤[‖K1‖Lp′ (Ω,ω) + C

p/p′

Ω ‖h1‖L∞(Ω)‖u‖p/p′

X

+ ‖h2‖L∞(Ω,ω)‖u‖p/p′

X + ‖u‖q/q′

X

]‖ϕ‖X .

Since B(u, ·) is linear, for each u ∈ X, there exists a linear and continuousoperator A : X → X∗ such that 〈Au,ϕ〉 = B(u, ϕ), for all u, ϕ ∈ X (where 〈f, x〉


denotes the value of the linear functional f at the point x) and

‖Au‖∗ ≤ ‖K1‖Lp′ (Ω,ω) + Cp/p′

Ω ‖h1‖L∞(Ω)‖u‖p/p′

X

+ ‖h2‖L∞(Ω,ω)‖u‖p/p′

X + ‖u‖q/q′

X .

Consequently, problem (1) is equivalent to the operator equation

Au = T, u ∈ X.

Step 4. Using condition (H2) and Lemma 1(b), we have

〈Au1 −Au2, u1 − u2〉 = B(u1, u1 − u2)−B(u2, u1 − u2)

=

∫Ω

ωA(x, u1,∇u1).∇(u1 − u2) dx+

∫Ω

|∆u1|q−2∆u1∆(u1 − u2)v dx

−∫

Ω

ωA(x, u2,∇u2) · ∇(u1 − u2) dx−∫

Ω

|∆u2|q−2∆u2∆(u1 − u2) v dx

=

∫Ω

ω(A(x, u1,∇u1)−A(x, u2,∇u2)

)· ∇(u1 − u2) dx

+

∫Ω

(|∆u1|q−2∆u1 − |∆u2|q−2∆u2)∆(u1 − u2)v dx

≥ θ1

∫Ω

ω|∇(u1 − u2)|p dx+ βq

∫Ω

(|∆u1|+ |∆u2|

)q−2|∆u1 −∆u2|2v dx

≥ θ1

∫Ω

ω |∇(u1 − u2)|p dx+ βq

∫Ω

(|∆u1 −∆u2|

)q−2|∆u1 −∆u2|2v dx

= θ1

∫Ω

ω|∇(u1 − u2)|p dx+ βq

∫Ω

|∆u1 −∆u2|qv dx

≥ 0.

Therefore, the operator A is monotone. Moreover, using (H3), we obtain

〈Au, u〉 = B(u, u) = B1(u, u) +B2(u, u)

=

∫Ω

ωA(x, u,∇u).∇udx+

∫Ω

|∆u|q−2∆u∆u v dx

≥∫

Ω

λ1|∇u|p ω dx+

∫Ω

|∆u|q v dx

= λ1 ‖∇u‖pLp(Ω,ω) + ‖∆u‖qLq(Ω,v).

Hence, since 1 < p, q <∞, we have

〈Au, u〉‖u‖X

→ +∞, as ‖u‖X → +∞ ,

(using limt+s→∞

tp + sq

t+ s=∞) that is, A is coercive.

Step 5. We need to show that the operator A is continuous.


Let um→u in X as m→∞. We have,

|B1(um, ϕ)−B1(u, ϕ)| ≤n∑j=1

∫Ω

|Aj(x, um,∇um)−Aj(x, u,∇u)||Djϕ|ω dx

=

n∑j=1

∫Ω

|Fjum − Fju||Djϕ|ω dx

≤n∑j=1

‖Fjum − Fju‖Lp′ (Ω,ω)‖Djϕ‖Lp(Ω,ω)

≤n∑j=1

‖Fjum − Fju‖Lp′ (Ω,ω)‖ϕ‖X ,

and

|B2(um, ϕ)−B2(u, ϕ)| =∣∣∣∣∫

Ω

|∆um|q−2∆um∆ϕv dx−

∫Ω

|∆u|q−2∆u∆ϕv dx

∣∣∣∣≤∫

Ω

∣∣∣|∆um|q−2∆um − |∆u|q−2∆u∣∣∣|∆ϕ|v dx

=

∫Ω

|Gum −Gu||∆ϕ|v dx

≤ ‖Gum −Gu‖Lq′ (Ω,v)‖∆ϕ‖Lq(Ω,v)

≤ ‖Gum −Gu‖Lq′ (Ω,v)‖ϕ‖X ,

for all ϕ ∈ X. Hence,

|B(um, ϕ)−B(u, ϕ)| ≤ |B1(um, ϕ)−B1(u, ϕ)|+ |B2(um, ϕ)−B2(u, ϕ)|

≤[ n∑j=1

‖Fjum − Fju‖Lp′ (Ω,ω) + ‖Gum −Gu‖Lq′ (Ω,v)

]‖ϕ‖X .

Then we obtain

‖Aum −Au‖∗ ≤n∑j=1

‖Fjum − Fju‖Lp′ (Ω,ω) + ‖Gum −Gu‖Lq′ (Ω,v) .

Therefore, using (4) and (5) we have ‖Aum −Au‖∗ → 0 as m→ +∞, that is, A iscontinuous (and this implies that A is hemicontinuous).

Therefore, by Theorem 4, the operator equation Au = T has a solution u ∈ Xand it is a solution for problem (1).Step 6. Let us now prove the uniqueness of the solution. Suppose that u1, u2 ∈ Xare two solutions of problem (1). Then,∫

Ω

|∆ui|q−2∆ui ∆ϕv dx+

n∑j=1

∫Ω

ωAj(x, ui(x),∇ui(x))Djϕdx

=

∫Ω

f0 ϕdx+

n∑j=1

∫Ω

fj Djϕdx,


for all ϕ ∈ X, and i = 1, 2. Hence, we obtain∫Ω

(|∆u1|q−2

∆u1 − |∆u2|q−2∆u2

)∆ϕv dx

+

∫Ω

ω(A(x, u1(x),∇u1(x))−A(x, u2(x),∇u2(x))

)· ∇ϕdx = 0 .

In particular, for ϕ = u1 − u2 ∈ X we have, by (H2) and Lemma 1(b) (analogousto Step 4),

0 =

∫Ω

ω(A(x, u1,∇u1)−A(x, u2,∇u2)

)· ∇(u1 − u2) dx

+

∫Ω

(|∆u1|q−2∆u1 − |∆u2|q−2

∆u2)∆(u1 − u2)v dx

≥ θ1

∫Ω

|∇(u1 − u2)|pω dx+ βq

∫Ω

|∆(u1 − u2)|qv dx .

Hence, ‖∇(u1 − u2)‖Lp(Ω,ω) = 0 and ‖∆(u1 − u2)‖Lq(Ω,v) = 0. Since u1, u2 ∈ X,then u1 = u2 µ1-a.e. Therefore, since ω ∈ Ap, we obtain that u1 = u2 a.e.

Example 1. Consider Ω = (x, y) ∈ R2 : x2 + y2 < 1, the weight functionsω(x, y) = (x2 + y2)−1/2 and v(x, y) = (x2 + y2)−2/3 (ω ∈ A3 and v ∈ A2, p = 3,q = 2), and the function

A : Ω× R2 → R2

A((x, y), η, ξ) = h2(x, y)|ξ|ξ,

where h(x, y) = 2 e(x2+y2). Let us consider the partial differential operator

Lu(x, y) = ∆((x2 + y2)−2/3|∆u|∆u

)− div

((x2 + y2)−1/2A((x, y), u,∇u)

).

Therefore, by Theorem 1, the problem (1)

Lu(x) =

cos(xy)

(x2 + y2)− ∂

∂x

(sin(xy)

(x2 + y2)

)− ∂

∂y

(sin(xy)

(x2 + y2)

), in Ω

u(x) = 0, on ∂Ω

∆u(x) = 0, on ∂Ω

has a unique solution u ∈ X = W 2,2(Ω, v) ∩W 1,30 (Ω, ω).

References

[1] A.C. Cavalheiro: Existence and uniqueness of solutions for some degenerate nonlinearDirichlet problems. J. Appl. Anal. 19 (2013) 41–54.

[2] A.C. Cavalheiro: Existence results for Dirichlet problems with degenerate p-Laplacian.Opuscula Math. 33 (3) (2013) 439–453.


[3] M. Chipot: Elliptic Equations: An Introductory Course. Birkhäuser, Berlin (2009).

[4] P. Drábek, A. Kufner, F. Nicolosi: Quasilinear Elliptic Equations with Degenerationsand Singularities. Walter de Gruyter, Berlin (1997).

[5] E. Fabes, C. Kenig, R. Serapioni: The local regularity of solutions of degenerate ellipticequations. Comm. Partial Differential Equations 7(1982) 77—116.

[6] S. Fučík, O. John, A. Kufner: Function Spaces. Noordhoff International Publ., Leiden(1977).

[7] J. Garcia-Cuerva, J.L. Rubio de Francia: Weighted Norm Inequalities and RelatedTopics. North-Holland Mathematics Studies (1985).

[8] D. Gilbarg, N.S. Trudinger: Elliptic Partial Equations of Second Order. Springer, NewYork (1983).

[9] J. Heinonen, T. Kilpeläinen, O. Martio: Nonlinear Potential Theory of DegenerateElliptic Equations. Oxford Math., Clarendon Press (1993).

[10] A. Kufner: Weighted Sobolev Spaces. John Wiley & Sons (1985).

[11] B. Muckenhoupt: Weighted norm inequalities for the Hardy maximal function. Trans.Amer. Math. Soc. 165 (1972) 207–226.

[12] M. Talbi, N. Tsouli: On the spectrum of the weighted p-Biharmonic operator withweight. Mediterr. J. Math. 4 (2007) 73–86.

[13] A. Torchinsky: Real-Variable Methods in Harmonic Analysis. Academic Press, SaoDiego (1986).

[14] B.O. Turesson: Nonlinear Potential Theory and Weighted Sobolev Spaces.Springer-Verlag (2000).

[15] E. Zeidler: Nonlinear Functional Analysis and its Applications, Vol. I. Springer-Verlag(1990).

[16] E. Zeidler: Nonlinear Functional Analysis and its Applications, Vol. II/B.Springer-Verlag (1990).

Author’s address:Department of Mathematics, State University of Londrina, 86057-970 Londrina,Brazil

E-mail: accava a©gmail.com

Received: 16th September, 2014Accepted for publication: 8th February, 2015Communicated by: Geoff Prince


On a class of nonlocal problem involving a criticalexponent

Anass Ourraoui

Abstract. In this work, by using the Mountain Pass Theorem, we give a re-sult on the existence of solutions concerning a class of nonlocal p-LaplacianDirichlet problems with a critical nonlinearity and small perturbation.

1 IntroductionThis paper deals with the following elliptic problem

−M(∫

Ω

|∇u|p dx

)∆pu = βh(x)|u|q−2u+ |u|p

∗−2u+ f(x) in Ω,

u = 0 on ∂Ω,

(1)

where Ω ⊂ RN is a bounded domain with smooth boundary, p∗ = NpN−p is the critical

Sobolev exponent, 1 < p < N , β is a positive parameter, and h ∈ Lp∗p∗−q (Ω), f ∈

Lp′(Ω), with 1

p + 1p′ = 1.

Where the functional M verifies,

M : (0,+∞)→ (0,+∞) is continuous and m0 = infs>0

M(s) > 0, (2)

The problem (1) is called nonlocal because of the presence of the termM(∫

Ω|∇u|p dx

), so it is not any more a pointwise identity. This leads us to

some mathematical difficulties which makes the study of such a class of problemparticularly interesting.

It is well known that the critical exponent case is often difficult because of thelack of compactness, so standard arguments cannot be carried out to handle theproblem (1). As far as we know, very few results have been obtained in elliptic

2010 MSC: 35J30, 35J60, 35J92Key words: p-Laplacian, Dirichlet problem, critical exponent.

48 Anass Ourraoui

problems involving critical exponent, for instance we just quote [1], [2], [4], [5],[6], [7], [9], [11] and references therein. However, inspired by these interestingworks, especially by [4], within which we will borrow some ideas, our goal willbe to generalize some corresponding results partially and extend them to the casep 6= 2 with an existence of a perturbation f. We have to mention that [5] couldbe considered as the first work dealing with multivalued elliptic problem and thepresence of which involves critical growth in an Orlicz-Sobolev space, where thenonlinearity can be discontinuous.

From now on, we make the following assumption:

M(t) ≥M(t)t for t > 0 , with M(t) =

∫ t

0

M(s) ds. (3)

Accordingly, we can report our main result,

Theorem 1. Under the hypotheses (2), (3) and q ∈ (p, p∗), there exists β∗ > 0,such that the problem (1) has at least a nontrivial solutions for all β ≥ β∗, providedf is small enough in the norm ‖·‖∗ of (W 1,p

0 (Ω))∗.

Throughout this paper, we consider the C1-functional energy

φ(u) =1

pM

(∫Ω

|∇u|p dx

)− β

q

∫Ω

h(x)|u|q dx− 1

p∗

∫Ω

|u|p∗

dx−∫

Ω

f(x)udx.

Note that

φ′(u) · v = M(‖u‖p)∫

Ω

|∇u|p−2∇u∇v dx− β∫

Ω

h(x)|u|q−2uv dx

−∫

Ω

|u|p∗−2uv dx−

∫Ω

f(x)v dx,

for all v ∈W 1,p0 (Ω). Where,

W 1,p0 (Ω) =

u ∈ Lp(Ω) :

∫Ω

|∇u|p dx <∞, u/∂Ω = 0.

By a version of the Mountain Pass Theorem due to Ambrosetti and Rabinowitz[10], [12], without Palais-Smale condition, there exists a sequence (un)n ⊂W 1,p

0 (Ω)such that

φ(un)→ cβ and φ′(un)→ 0,

where

cβ = infγ∈Γ

maxt∈[0,1]

φ(γ(t)) > 0

with

Γ =γ ∈ C

([0, 1],W 1,p

0 (Ω))

: γ(0) = 0, φ(γ(1)

)< 0.

On a class of nonlocal problem involving a critical exponent 49

We recall that u ∈W 1,p0 (Ω) is a weak solution of the problem (1) if it verifies

M(‖u‖p)∫

Ω

|∇u|p−2∇u∇v dx−∫

Ω

βh(x)|u|q−2uv dx

−∫

Ω

|u|p∗−2uv dx−

∫Ω

f(x)v dx = 0,

for all v ∈W 1,p0 (Ω).

So the critical points of φ are solutions of the problem (1).

2 Auxiliary resultsLet Ls(Ω) be the Lebesgue space equipped with the norm |u|s =

(∫Ω|u|s dx

) 1s ,

1 ≤ s <∞ and let W 1,p0 (Ω) be the usual Sobolev space with respect to the norm

‖u‖ =

(∫Ω

|∇u|p dx

) 1p

.

Now we can define the best Sobolev constant

S = infu∈W 1,p(Ω)\0

∫Ω|∇u|pdx

(∫

Ω|u|p∗dx)

pp∗.

In the sequel, we are to compare the minimax level cβ with a suitable numberwhich involves the constant S.

Lemma 1. There exist σ > 0, ρ > 0 and e ∈W 1,p0 (Ω) with ‖e‖ > ρ such that

(i) inf‖u‖=ρ φ(u) ≥ σ > 0;

(ii) φ(e) < 0.

Proof. (i) From the Hölder’s inequality and the compact embedding theorem, wehave

φ(u) ≥ m0

p

∫Ω

|∇u|p dx− β

q|h|θ

∫Ω

|u|q dx− 1

p∗

∫Ω

|u|p∗

dx−∫

Ω

f(x)udx

≥ C0‖u‖p −C1β

q|h|θ‖u‖q −

1

p∗Sp∗p

‖u‖p∗− |f |p′ |u|p

≥ C0‖u‖p −C1β

q|h|θ‖u‖q − C2‖u‖p

∗− C3‖f‖∗‖u‖, (4)

with θ = p∗

[p∗−q] and C0, C1, C2, C3 > 0. Since q ∈ (p, p∗) then for ‖u‖ = ρ > 0

small enough, we may find σ > 0 such that

inf‖u‖=ρ

φ(u) ≥ σ > 0

where ‖f‖∗ be small.

50 Anass Ourraoui

(ii) Fix v ∈ C∞0 (Ω \ 0) with v ≥ 0 in Ω and ‖v‖ = 1.

φ(tv) ≤ A|t|p − β|t|θ∫

Ω

h(x)vθ dx+ C − |t|p∗

p∗

∫Ω

h(x)vp∗

dx− |t|∫

Ω

f(x)v dx,

with A and C are two positive constants, it follows that

φ(tv)→ −∞ as |t| → ∞.

Lemma 2. limβ→+∞

cβ = 0.

Proof. Let v the function given by the previous lemma 1, then there is tβ > 0 suchthat φ(tβv) = max

t≥0φ(tv), thereafter,

M(‖tβv‖p)tpβ‖v‖p = βtqβ

∫Ω

h(x)|v|q dx+ tp∗

β

∫Ω

|v|p∗

dx+ t2β

∫Ω

f(x)v2 dx, (5)

it follows from (3) that there is c > 0, such that

M(s) ≤ c|s| for all s > s0 > 0 .

Hence

ctpβ‖v‖p ≥ βtqβ

∫Ω

h(x)|v|q dx+ tp∗

β

∫Ω

|v|p∗

dx+ t2β

∫Ω

f(x)v2 dx

and then tβ is bounded, so there exists a sequence βn → +∞ and t∗ ≥ 0 withtβn → t∗ as n→ +∞ and thus

M(‖tβnv‖p)tpβn‖v‖p < C, ∀n ∈ N,

with C is a positive constant, which yields

βntq∗

∫Ω

h(x)|v|q dx+ tp∗

∗

∫Ω

|v|p∗

dx ≤ C, ∀n ∈ N.

Hence, we claim that t∗ = 0, otherwise, t∗ > 0 and then the last inequality becomes

βntq∗

∫Ω

h(x)|v|q dx+ tp∗

∗

∫Ω

|v|p∗

dx→ +∞

as n→ +∞, which is absurd, so t∗ = 0.Taking γ0(t) = te, with γ0 ∈ Γ, then we get

0 < cβ ≤ maxt∈[0,1]

φ(γ0(t)) ≤ 1

pM(tpβ).

Since M(tpβ)→ 0 then limβ→∞ cβ = 0.


As consequence of the above lemma, there exists β∗ > 0 such that for everyβ ≥ β∗,

cβ <

(1− p

p∗

)(m0S)

Np .

Lemma 3. Let (un)n ⊂ W 1,p0 (Ω), with φ(un) → cβ , and φ′(un) → 0. Then (un)n

is bounded in W 1,p0 (Ω).

Proof. Assume that φ(un)→ cβ , and φ′(un)→ 0, then we have

pcβ + o(1) + o(1)‖un‖ = pφ(un)− (φ′(un) · un)

≥ C4β

(1− p

q

)|h|θ‖un‖q + C5

(1− p

p∗

)‖un‖p

∗

+ (p− 1)

∫Ω

f(x)un dx,

where θ = p∗

p∗−q , C4, C5 > 0, we infer that (un)n is bounded in W 1,p0 (Ω).

3 Proof of the main resultProof. (Theorem 1) As it was previously mentioned, we are to apply a version ofthe Mountain Pass theorem without Palais-Smale condition to obtain a sequence(un)n ⊂W 1,p

0 (Ω) such that φ(un)→ cβ and φ′(un)→ 0.

Because (un)n is a bounded sequence in W 1,p0 (Ω), passing to a subsequence, so

we may find γ > 0 with‖un‖ → γ,

it follows from the continuity of M that

M(‖un‖p)→M(γp).

On the other side, we know that un u in W 1,p0 (Ω), then

un → u in Lr(Ω), for 1 < r < p∗

andun(x)→ u(x) a.e. x ∈ Ω.

By the Lebesgue Dominated Theorem,∫Ω

h(x)|un|q dx→∫

Ω

h(x)|u|q dx.

Further,|∇un|p |∇u|p + µ weak∗-sense of measure,

|un|p∗ |u|p

∗+ ν weak∗-sense of measure.

Afterwards, as a consequence of the concentration compactness principle dueto Lion [8], there is an index set I, which is an at most countable set such that

ν =∑i∈I

νiδi, µ ≥∑i∈I

µiδi

52 Anass Ourraoui

andSν

p/p∗

i ≤ µi,

for any i ∈ I with (µi)i, (νi)i ⊂ [0,∞), δi is the Dirac mass and (µi)i, (νi)i arenonatomic positive measures. We claim that I = ∅, otherwise, we have I 6= ∅ andfix i ∈ I. Taking ψ ∈ C∞0 (Ω, [0, 1]) such that ψ ≡ 1 if |x| < 1 and ψ ≡ 0 when|x| > 2 with |∇ψ|∞ ≤ 2. Putting ψρ(x) = ψ

(x−xiρ

)for ρ > 0, noting that (ψρun)

is bounded thus φ′(un) · (ψρun)→ 0, that is

M

(∫Ω

|∇un|p)∫

Ω

|∇un |p−2 ∇un · ∇ψρun dx

= −M(∫

Ω

|∇un|p)∫

Ω

|∇un |p ψρ∇un dx+

∫Ω

|un|p∗−2un · ψρun dx

+ β

∫Ω

h(x)|un|q−2unψρun dx+

∫Ω

f(x)ψρun +On(1).

As it is known that B2ρ(xi) is the support of the functional ψρ and by applyingHölder inequality then we get∣∣∣∫

Ω

|∇un|p−2∇un.∇ψρun dx∣∣∣ ≤ ∫

B2ρ(xi)

|∇un|p−1 |un∇ψρ|dx

≤(∫

B2ρ(xi)

|∇un|p) 1p′(∫

B2ρ(xi)

|un∇ψρ|p dx) 1p

≤ C(∫

B2ρ(xi)

|un∇ψρ|p dx) 1p

.

By the Dominated convergence Theorem we entail that∫B2ρ(xi)

|un∇ψρ|p dx→ 0

when n→∞ and ρ→ 0.Hence,

limρ→0

[limn

∫Ω

un|∇un|p−2∇un.∇ψρ]

= 0.

On the other hand, we recall that M(‖un‖p) converges to M(γp), so we reach

limρ→0

[limnM(‖un‖p)

∫Ω

un|∇un|p−2∇un.∇ψρ]

= 0.

Similarly,

limρ→0

limn

[∫Ω

h(x)|un|q−2unψρun

]= 0,

limρ→0

limn

[∫Ω

f(x)ψρun

]= 0


and thus ∫Ω

M(γp)ψρ dµ+Oρ(1) ≤∫

Ω

ψρ dν.

Tending ρ to zero we conclude that

νi ≥M(γP )µi ≥ m0µi,

from the definition of ν and µ we have

νi ≥ (m0S)Np .

It does not make sense, indeed, let i ∈ I such that

νi ≥ (m0S)Np .

Since (un)n is a (PS)cβ for the functional φ, then

pcβ = pφ(un) = pφ(un)− φ′(un) · un +On(1)

≥(

1− p

p∗

)∫Ω

ψρ|un|p∗

dx+On(1),

tending n→ +∞, therefore

pcβ ≥(

1− p

p∗

)∑i∈I

ψρ(xi)νi =

(1− p

p∗

)∑i∈I

νi ≥(

1− p

p∗

)(m0S)

Np ,

which cannot occur (because limβ→∞ cβ = 0), thereafter I is empty and therebyun → u in Lp

∗(Ω).

On the other hand,

M (‖un‖p)∫

Ω

(|∇un|p−2∇un − |∇u|p−2∇u

)(∇un −∇u) dx

= φ′(un). (un − u) + β

∫Ω

h(x)|un|q−2un(un − u) dx+

∫Ω

f(x)(un − u) dx

+

∫Ω

|un|p∗−2un(un − u) dx−M (‖un‖p)

∫Ω

|∇u|p−2∇u (∇un −∇u) dx.

In view of un u, a standard argument (similar to those found in [3]) showsthat

∇un(x)→ ∇u(x) for a.e. x ∈ Ω,

andun(x)→ u(x) for a.e. x ∈ Ω,

then

M (‖un‖p)∫

Ω

(|∇un|p−2∇un − |∇u|p−2∇u

)(∇un −∇u) dx→ 0.

54 Anass Ourraoui

Using the following inequalities ∀x, y ∈ RN

|x− y|γ ≤ 2γ(|x|γ−2x− |y|γ−2y) · (x− y) if γ ≥ 2,

|x− y|2 ≤ 1

γ − 1(|x|+ |y|)2−γ(|x|γ−2x− |y|γ−2y) · (x− y) if 1 < γ < 2,

where x · y is the inner product in RN , we get

c m0

∫Ω

|∇un−∇u|p dx ≤M (‖un‖p)∫

Ω

(|∇un|p−2∇un − |∇u|p−2∇u

)(∇un −∇u) dx.

Consequently,‖un − u‖ → 0,

which will imply thatun → u in W 1,p

0 (Ω).

Thusφ(u) = cβ , φ′(u) = 0

and we get the solution u1, it is a mountain pass type.

AcknowledgementsThe referee deserves many thanks for careful reading and many useful comments.

References

[1] C.O. Alves, F.J.S.A. Correa, G.M. Figueiredo: On a class of nonlocal elliptic problemswith critical growth. Differential Equation and Applications 2 (2010) 409–417.

[2] J.G. Azorero, I.P. Alonso: Multiplicity of solutions for elliptic problems with criticalexponent or with a nonsymmetric term. Trans. Amer. Math. Soc. 323 (2) (1991)877–895.

[3] A. El Hamidi, J.M. Rakotoson: Compactness and quasilinear problems with criticalexponents. Differ. Integral Equ. 18 (2005) 1201–1220.

[4] M. G. Figueiredo: Existence of a positive solution for a Kirchhoff problem type withcritical growth via truncation argument. J. Math. Anal. Appl. 401 (2013) 706–713.

[5] G. M. Figueiredo, Jefferson A. Santos: On a Φ-Kirchhoff multivalued problem withcritical growth in an Orlicz-Sobolev space. Asymptotic Analysis 89 (1) (2014) 151–172.

[6] A. Fiscella, E. Valdinoci: A critical Kirchhoff type problem involving a nonlocaloperator. Nonlinear Anal. 94 (2014) 156–170.

[7] N. Fukagai, K. Narukawa: Positive solutions of quasilinear elliptic equations with criticalOrlicz–Sobolev nonlinearity on RN. Funkciallaj Ekvacioj 49 (1981) 235–267.

[8] P. L. Lions: The concentraction-compactness principle in the calculus of virations. Thelimit case, Part 1. Rev Mat Iberoamericana 1 (1985) 145–201.

[9] A. Ourraoui: On a p-Kirchhoff problem involving a critical nonlinearity. C. R. Acad. Sci.Paris 352 (2014) 295–298.

[10] P. Pucci: Geometric description of the mountain pass critical points. ContemporaryMathematicians 2 (2014) 469–471.


[11] P. Pucci, S. Saldi: Critical stationary Kirchhoff equations in RN involving nonlocaloperators. Rev. Mat. Iberoam.(2014).

[12] M. Willem: Minimax Theorems. Springer (1996).

Author’s address:ENSAH, University of Mohamed I, Oujda, Morocco

E-mail: a.ourraoui a©gmail.com

Received: 14th October, 2014Accepted for publication: 27th April, 2015Communicated by: Olga Rossi


Newton transformations on null hypersurfaces

Cyriaque Atindogbé and Hans Tetsing Fotsing

Abstract. Any rigged null hypersurface is provided with two shape opera-tors: with respect to the rigging and the rigged vector fields respectively.The present paper deals with the Newton transformations built on both ofthem and establishes related curvature properties. The laters are used toderive necessary and sufficient conditions for higher-order umbilicity andmaximality we introduced in passing, and develop general Minkowski-typeformulas for the null hypersurface, supported by some physical models inperfect-fluid space-times.

1 IntroductionIt is a well-known fact that null hypersurfaces are exclusive objects of pseudo--Riemannian manifolds in the sense that they have no Riemannian counterpartand hence are interesting on their own from a (differential) geometric point ofview. They also play an important role in general relativity namely in the study ofblack hole horizons (regions of space-time which contains a huge amount of masscompacted into an extremely small volume). From a more technical aspect, theyare hypersurfaces having (induced) metrics with (pointwise) vanishing determi-nants and this degeneracy leads to several difficulties. In pseudo-Riemannian case,due to the causal character of three categories of vector fields (namely, spacelike,timelike and null), the induced metric on a hypersurface is a non-degenerate metrictensor field or a degenerate symmetric tensor field depending on whether the normalvector field is of the first two types or the third one. On non-degenerate hypersur-faces one can consider all the fundamental intrinsic and extrinsic geometric notions.In particular, a well defined (up to sign) notion of the unit orthogonal vector field isknown to lead to a canonical splitting of the ambient tangent space into two factors:a tangent and an orthogonal one. Therefore, by respective projections, one has fun-damental equations such as the Gauss, the Codazzi, the Weingarten equations, . . .along with the second fundamental form, shape operator, induced connection, etc.

2010 MSC: 53C42, 53B30, 53Z05Key words: Null hypersurfaces, null rigging, Newton transformations, Minkowski integral

formulas.

58 Cyriaque Atindogbé and Hans Tetsing Fotsing

The null hypersurface case is precisely when the normal vector field is null (alsocalled lightlike) and since (contrary to the non-degenerate counterpart) the nor-mal vector bundle intersects (non trivially) with the tangent bundle, one cannotfind natural projector (and hence there is no preferred induced connection such asLevi-Civita) to define induced geometric objects as usual. This degeneracy of theinduced metric makes it impossible to study them as part of standard submanifoldtheory, forcing to develop specific techniques and tools. For the most part, thesetools are specific to a given problem, or sometimes with auxiliary non-canonicalchoices on which, unfortunately, depends the constructed null geometry. Indeed,Duggal and Bejancu in [12] introduced a non-degenerate screen distribution (orequivalently a null transversal line vector bundle as we may see below) so as to geta three factors splitting of the ambient tangent space and derive the main inducedgeometric objects such as second fundamental forms, shape operators, induced con-nections, curvature, etc. Unfortunately, the screen distribution is not unique andthere is no preferred one in general, unless some specific geometric conditions areformulated to select and ensure uniqueness in exceptional cases [6], [5], [8], [7].From above mentioned difficulties and compared to extensive research on globalRiemannian and Lorentzian geometries we find out that considerable works areneeded in null geometry to fill the gap.

One of the most important and central tools which have been extremely usefulin addressing issues on higher-order r-th mean curvature and related topics in Rie-mannian geometry are Newton transformations [1], [2], [3], [4], [10], [15]. Since anynull hypersurface with a fixed rigging do carry two shape operators: with respectto the rigging and the rigged vector fields respectively, we reasonably expect a roleof those transformations in the study of null hypersurfaces. Recently in [11], theauthors used above transformations of first type (thus, by duality considering thescreen structure but not the null hypersurface structure) and examine conditionsunder which compact null hypersurfaces are totally umbilical in Robertson-Walker(RW) space-times. In the present paper we consider Newton transformations builton both of the two shape operators and establish related curvature properties andderive necessary and sufficient conditions for higher-order umbilicity and maximal-ity, along with general Minkowski-type formulas for null hypersurfaces. The paperis organized as follows. Section 2 sets notations and definitions on riggings (nor-malizations) and review basic properties on null hypersurfaces, followed by sometechnical lemmas. Section 3 starts with introducing Newton transformations withrespect to the rigged vector field and establishes their basic properties and somecharacterization results. The behaviour with respect to change in rigging is thenexamined on these transformations and the section ends with establishing someMinkowski-type integral formulas. In Section 4 we present some physical mod-els in perfect-fluid space-times. The last section is concerned with the Newtontransformations with respect to the rigging vector field.

2 PreliminariesLet (M, g) be an (n+ 2)-dimensional Lorentzian manifold and M a null hypersur-face in M . This means that at each p ∈ M , the restriction gp|TpM is degenerate,that is there exists a non-zero vector U ∈ TpM such that g(U,X) = 0 for all

Newton transformations on null hypersurfaces 59

X ∈ TpM . Hence, in null setting, the normal bundle TM⊥ of the null hypersur-face Mn+1 is a rank 1 vector subbundle of the tangent bundle TM , contrary tothe classical theory of non-degenerate hypersurfaces for which the normal bundlehas trivial intersection 0 with the tangent one and plays an important role inthe introduction of the main induced geometric objects on M . Let us start withthe usual tools involved in the study of such hypersurfaces according to [12]. Theyconsist in fixing on the null hypersurface a geometric data formed by a lightlikesection and a screen distribution. By screen distribution on Mn+1, we mean acomplementary bundle of TM⊥ in TM . It is then a rank n non-degenerate distri-bution over M . In fact, there are infinitely many possibilities of choices for sucha distribution provided the hypersurface M be paracompact, but each of them iscanonically isomorphic to the factor vector bundle TM/TM⊥. For reasons thatwill become obvious in few lines below, let denote such a distribution by S (N).We then have

TM = S (N)⊕Orth TM⊥, (1)

where ⊕Orth denotes the orthogonal direct sum. From [12], it is known that for anull hypersurface equipped with a screen distribution, there exists a unique rank 1vector subbundle tr(TM) of TM over M , such that for any non-zero section ξ ofTM⊥ on a coordinate neighbourhood U ⊂ M , there exists a unique section N oftr(TM) on U satisfying

g(N, ξ) = 1, g(N,N) = g(N,W ) = 0, ∀W ∈ S (N)|U . (2)

Then TM is decomposed as follows:

TM |M = TM ⊕ tr(TM) = TM⊥ ⊕ tr(TM) ⊕Orth S (N). (3)

We call tr(TM) a (null) transversal vector bundle along M . In fact, from (2) and(3) one shows that, conversely, a choice of a transversal bundle tr(TM) determinesuniquely the screen distribution S (N). A vector field N as in (2) is called anull transversal vector field of M . It is then noteworthy that the choice of a nulltransversal vector field N along M determines both the null transversal vectorbundle, the screen distribution S (N) and a unique radical vector field, say ξ,satisfying (2). Tangent vector fields to S (N) (resp. to TM⊥) are called horizontal(resp. vertical). Now, to continue our discussion, we need to clarify the concept ofrigging for our null hypersurface.

Definition 1. Let M be a null hypersurface of a Lorentzian manifold. A riggingforM is a vector field L defined on some open set containingM such that Lp /∈ TpMfor each p ∈M .

An outstanding property of a rigging is that it allows definition of geometricobjects globally onM . We say that we have a null rigging in case the restriction of Lto the null hypersurface is a null vector field. From now on we fix a null rigging Nfor M . In particular this rigging fixes a unique null vector field ξ ∈ Γ(TM⊥)called the rigged vector field, all of them defined in an open set containing M(hence globally on M) such that (1), (2) and (3) hold. Whence, from now on,


by a normalized (or rigged) null hypersurface we mean a triplet (M, g,N) whereg = g|M is the induced metric on M and N a null rigging for M . In fact, incase the ambient manifold M has Lorentzian signature, at an arbitrary point pin M , a real null cone Cp is invariantly defined in the (ambient) tangent spaceTpM and is tangent to M along a generator emanating from p. This generatoris exactly the radical fibre ∆p = TpM

⊥ and for each null rigging N for M andeach p ∈ M we have Np ∈ Cp \ ∆p. Actually, a lightlike hypersurface M of aLorentzian manifold is a hypersurface which is tangent to the lightlike cone Cpat each point p ∈ M . Recall that a space-time (M, g) is a connected Lorentzianmanifold which is “time-oriented”, i.e. a causal cone at each TpM , p ∈ M (the“future” causal cone) has been continuously chosen. Hence, null hypersurfaces inspace-times can be naturally given an orientation by such a continuous districutionof causal cones Cp.

Let N be a null rigging of a null hypersurface of a Lorentzian manifold (M, g)and θ = g(N, ·) the 1-form metrically equivalent to N defined on M . Then, take

η = i?θ

to be its restriction to M , the map i : M → M being the inclusion map. Thenormalization (M, g,N) will be said to be closed if the 1-form η is closed on M .It is easy to check that S (N) = ker(η) and that the screen distribution S (N) isintegrable whenever η is closed. On a normalized null hypersurface (M, g,N), theGauss and Weingarten formulas are given by

∇XY = ∇XY +BN (X,Y )N,

∇XN = −ANX + τN (X)N,

∇XPY =?∇XPY + CN (X,PY )ξ,

∇Xξ = −?AξX − τN (X)ξ,

for any X,Y ∈ Γ(TM), where ∇ denotes the Levi-Civita connection on (M, g),∇ denotes the connection on M induced from ∇ through the projection along

the rigging N and?∇ denotes the connection on the screen distribution S (N)

induced from ∇ through the projection morphism P of Γ(TM) onto Γ(S (N)

)with respect to the decomposition (1). Now the (0, 2) tensors BN and CN are

the second fundamental forms on TM and S (N) respectively, AN and?Aξ are the

shape operators on TM and S (N) respectively and τN a 1-form on TM definedby

τN (X) = g(∇XN, ξ).

For the second fundamental forms BN and CN the following holds

BN (X,Y ) = g(?AξX,Y ), CN (X,PY ) = g(ANX,Y ) ∀X,Y ∈ Γ(TM), (4)

andBN (X, ξ) = 0,

?Aξξ = 0. (5)


It follows from (5) that integral curves of ξ are pregeodesics in both M and M , as∇ξξ = ∇ξξ = −τN (ξ)ξ. Throughout the paper, and without explicit mention, weconsider these integral curves to be geodesics which means that

τN (ξ) = 0.

A null hypersurface M is said to be totally umbilical (resp. totally geodesic)if there exists a smooth function ρ on M such that at each p ∈ M and for allu, v ∈ TpM , BN (p)(u, v) = ρ(p)g(u, v) (resp. BN vanishes identically on M). Theseare intrinsic notions on any null hypersurface in the following way. Note that Nbeing a null rigging for M , a vector field N ∈ Γ(TM) is a null rigging for M if andonly if it is defined in an open set containing M and there exist a function ψ on Mand a section ζ of TM such that Ni = (ψN)i+ζ with the properties that φ = ψiis nowhere vanishing, being i the inclusion map, and 2φη(ζ) + ‖ζ‖2 = 0 along M .

Then we have (see [7] for details on changes in normalizations) BN = 1ψiB

N whichshows that total umbilicity and total geodesibility are intrinsic properties for M .The total umbilicity and the total geodesibility conditions for M can also be written

respectively as?Aξ = ρP and

?Aξ = 0. Also, the screen distribution S (N) is totally

umbilical (resp. totally geodesic) if CN (X,PY ) = λg(X,Y ) for all X,Y ∈ Γ(TM)(resp. CN = 0), which is equivalent to AN = λP (resp. AN = 0). It is noteworthy

to mention that the shape operators?Aξ and AN are S (N)-valued.

The induced connection ∇ is torsion-free, but not necessarily g-metric unless Mis totally geodesic. In fact we have for all tangent vector fields X,Y and Z in TM ,

(∇Xg)(Y,Z) = BN (X,Y )η(Z) +BN (X,Z)η(Y ). (6)

Denote by R and R the Riemann curvature tensors of ∇ and ∇, respectively.Then the following are the Gauss-Codazzi equations [12, p. 93].⟨

R(X,Y )Z, ξ⟩

= (∇XBN )(Y,Z)− (∇YBN )(X,Z)

+ τN (X)BN (Y, Z)− τN (Y )BN (X,Z), (7)⟨R(X,Y )Z,PW

⟩=⟨R(X,Y )Z,PW

⟩+BN (X,Z)CN (Y, PW )

−BN (Y,Z)CN (X,PW ),⟨R(X,Y )ξ,N

⟩=⟨R(X,Y )ξ,N

⟩= CN (Y,

?AξX)− CN (X,

?AξY )

− 2dτN (X,Y ),⟨R(X,Y )PZ,N

⟩=⟨(∇XAN )Y, PZ

⟩−⟨(∇YAN )X,PZ

⟩+ τN (Y )

⟨ANX,PZ

⟩− τN (X)

⟨ANY, PZ

⟩(8)

for all X,Y, Z,W ∈ Γ(TM |U ). The (shape) operator?Aξ is self-adjoint as the

second fundamental form BN is symmetric. However, this is not the case for theoperator AN as shown in the following lemma.

Lemma 1. For all X,Y ∈ Γ(TM),⟨ANX,Y

⟩−⟨ANY,X

⟩= τN (X)η(Y )− τN (Y )η(X)− 2dη(X,Y ),

where (throughout)⟨·, ·⟩

= g stands for the Lorentzian metric.


Proof. Recall that η = i?θ where θ = 〈N, ·〉. Taking the differential of θ and usingthe Weingarten formula, we have for all X,Y ∈ Γ(TM),

2dη(X,Y ) = 2dθ(X,Y ) =⟨∇XN,Y

⟩−⟨∇YN,X

⟩= −

⟨ANX,Y

⟩+ τN (X)η(Y ) +

⟨ANY,X

⟩− τN (Y )η(X).

Hence, ⟨ANX,Y

⟩−⟨ANY,X

⟩= τN (X)η(Y )− τN (Y )η(X)− 2dη(X,Y )

as announced.

In case the normalization is closed the (connection) 1-form τN is related to theshape operator of M as follows.

Lemma 2. Let (M, g,N) be a closed normalization of a null hypersurface M in aLorentzian manifold such that τN (ξ) = 0. Then

τN = −⟨ANξ, ·

⟩.

Proof. Assume η = i?θ closed and let X, Y be tangent vector fields to M . The con-dition X ·η(Y )−Y ·η(X)−η([X,Y ]) = 0 is equivalent to

⟨∇XN,Y

⟩=⟨∇YN,X

⟩.

Then by the Weingarten formula, we get⟨−ANX,Y

⟩+ τN (X)η(Y ) =

⟨−ANY,X

⟩+ τN (Y )η(X).

In this relation, take Y = ξ to get

τN (X) = −⟨ANξ,X

⟩+ τN (ξ)η(X)

which gives the desired formula as τN (ξ) = 0.

The following relations (see a detailed proof in [6]) account for effects of therigging change N −→ N|M = φN + ζ on the induced geometric objects described

in Section 2. Throughout, items with the symbol ∼ apply to N .

ξ =1

φξ, BN (X,Y ) =

1

φBN (X,Y ), P = P − 1

φg(ζ, ·)ξ

CN (X, PY ) = φCN (X,PY )− g(∇Xζ, PY )

+

[τN (X) +

X · φφ

+1

φBN (ζ,X)

]g(ζ, Y )

(9)

∇XY = ∇XY −1

φBN (X,Y )ζ,

?Aξ =

1

φ

?Aξ −

1

φ2BN (ζ, ·)ξ (10)

AN = φAN −∇.ζ +

[τN + d ln|φ|+ 1

φBN (ζ, ·)

]ζ

for all tangent vector fields X and Y . Throughout the following ranges of indicesis used: i, j, l = 1, . . . , n, α, β = 0, 1, . . . , n, a, b = 0, 1, . . . , n+ 1.


3 Newton transformations and Minkowski integral formulas withrespect to the rigged section

Due to the first relation in (4), it is noteworthy that among the two shape operators

carried out by the rigged null hypersurface M ,?Aξ is actually the one that encodes

at best its null geometry. We introduce in this section the Newton transformationscorresponding to it. The second one AN is instead more concerned with the screenstructure S (N) and will be considered subsequently.

3.1 Newton transformations of?Aξ

Let (M, g,N) be an (n+ 1)-dimensional normalized null hypersurface with rigged

vector field ξ. Relation (4) shows that?Aξ is a self-adjoint linear operator on each

fibre TpM (p ∈ M) and?Aξξ = 0. Then,

?Aξ is diagonalizable and have (n + 1)

real-valued eigenfunctions?k0 = 0,

?k1, . . . ,

?kn called principal curvatures of the

null hypersurface with respect the shape operator?Aξ. With respect to a quasi-

orthonormal frame field ?E0 = ξ,

?E1, . . . ,

?En of corresponding eigenvector fields

the matrix of?Aξ take the form

0 0 · · · 0

0?k1 · · · 0

......

. . ....

0 0 · · ·?kn

.

The function?H1 = 1

n+1 tr(?Aξ) is the mean curvature function of the null hyper-

surface and is a member of a familly of n + 1 similar invariants (?Hr)0≤r≤n called

r-th mean curvature given by

?Hr =

(n+ 1

r

)−1

σr(?k0, . . . ,

?kn) and

?H0 = 1 (constant function 1),

where for 1 ≤ r ≤ n, the algebraic invariant σr is the r-th elementary symmetricpolynomial given by

σr(?k0, . . . ,

?kn) =

∑0≤i1<···<ir≤n

?ki1 · · ·

?kir .

It follows that the characteristic polynomial of?Aξ is given by

P (t) = det(?Aξ − tI) =

n+1∑a=0

(−1)a(n+ 1

a

)?Hrt

n+1−a.

Set?Sr = σr(

?k0, . . . ,

?kn) and

?Sαr = σr(

?k0, . . . ,

?kα−1,

?kα+1, . . . ,

?kn).


Definition 2. Let r be an integer such that 1 ≤ r ≤ n. The null hypersurface Mis r-umbilical (resp. r-maximal) if

?Sir =

?Sjr ∀i, j ∈ 1, . . . , n (resp.

?Hr = 0).

Remark 1. 1. As we show below (18) both r-maximality and r-total umbilicityare independent of the rigging.

2. The r-total umbilicity (respectively, r-maximality) generalize the totally um-bilical (respectively, maximal) obtained when r = 1. But, it is easy to checkthat any totally umbilical hypersurface is r-totally umbilical for all r.

3. For a 4-dimensional null hypersurface (i.e. n = 3), total umbilicity and 2-totalumbilicity are equivalent.

Example 1.Consider the 6-dimensional space M = R6 endowed with the Lorentzianmetric

g = −(dx0)2 + (dx1)2 + exp 2x0[(dx2)2 + (dx3)2] + exp 2x1[(dx4)2 + (dx5)2],

(x0, . . . , x5) being the usual rectangular coordinates on M . The only non-zeroChristoffel coefficients of the Levi-Civita connection of g are

Γ202 = Γ3

03 = Γ414 = Γ5

15 = 1, Γ022 = Γ0

33 = − exp 2x0, Γ144 = Γ1

55 = exp 2x1.

Now, consider the hypersurface M of M define by

M = (x0, . . . , x5) ∈ R6 ; x0 + x1 = 0.

Then, M is a null hypersurface of (M, g) and the vector field N = − 12

(∂∂x0 + ∂

∂x1

)is a null rigging for M with rigged vector field ξ = ∂

∂x0 − ∂∂x1 and we have

S (N) = span?E1,

?E2,

?E3,

?E4 with

?E1 = e−2x0 ∂

∂x2,

?E2 = e−2x0 ∂

∂x3,

?E3 = e−2x1 ∂

∂x4,

?E4 = e−2x1 ∂

∂x5.

Then it is easy to check that

∇ ?E1ξ =

?E1 ⇒

?k1 = −1,

∇ ?E2ξ =

?E2 ⇒

?k2 = −1,

∇ ?E3ξ = −

?E3 ⇒

?k3 = 1,

∇ ?E4ξ = −

?E4 ⇒

?k4 = 1.

Hence, M is 2-totally umbilical but it is not totally umbilical.


For each r = 0, . . . , n+1, the r-th Newton transformation?Tr : Γ(TM)→ Γ(TM)

of the endomorphism?Aξ, is given by

?Tr =

r∑a=0

(−1)a?Sa

?Ar−aξ .

Inductively,?T0 = I and

?Tr = (−1)r

?SrI +

?Aξ

?Tr−1,

where I denotes the identity map in Γ(TM). According to the Cayley-Hamilton

theorem, we have?Tn+1 = 0. By elementary algebraic computations, the following

is straightforward.

Proposition 1. (a)?Tr is self-adjoint and commute with

?Aξ;

(b)?Tr

?Eα = (−1)r

?Sαr

?Eα;

(c) tr(?Tr) = (−1)r(n+ 1− r)

?Sr;

(d) tr( ?Aξ

?Tr−1

)= (−1)r−1r

?Sr;

(e) tr( ?A2ξ

?Tr−1

)= (−1)r

( ?S1

?Sr + (r + 1)

?Sr+1

);

(f) tr(?Tr−1 ∇X

?Aξ) = (−1)rX ·

?Sr.

Proof. The first item is due to the fact that?Aξ is self-adjoint. We show (b) induc-

tively. In item (b) observe that the equality is trivial for r = 0. Assume that (b)

holds for r− 1 and observe that?Sαr =

?Sr −

?kα

?Sαr−1. Then using the above and the

well-known iterative relation characterizing the?Tr, we get,

?Tr

?Eα = (−1)r

?Sr

?Eα +

?Aξ

?Tr−1

?Eα

= (−1)r( ?Sr −

?kα

?Sαr−1

) ?Eα

= (−1)r?Sαr

?Eα

which shows (b). Through the above proof of b we see that (−1)r?Sαr are eigenfunc-

tions associated to?Eα for each α and then we have

tr(?Tr) = (−1)r

n∑α=0

?Sαr

and each of the(n+1r

)degree r monomials of

?Sr can be counted (n + 1)

(nr

)times

in the above summation. Thusn∑α=0

?Sαr =

(n+ 1)(nr

)(n+1r

) ?Sr = (n+ 1− r)

?Sr


and (c) is proved. By using the iterative formula of?Tr,

tr( ?Aξ

?Tr−1

)= tr(

?Tr) + (−1)r

?Sr tr(I)

= (−1)r(

(n+ 1− r)?Sr + (n+ 1)

?Sr

)= (−1)r−1r

?Sr,

that is (d). Item (e) is immediate as

tr( ?A2ξ

?Tr−1

)= tr(

?Aξ

?Tr) + (−1)r

?Sr tr(

?Aξ)

= (−1)r( ?S1

?Sr + (r + 1)

?Sr+1

).

Finally,

g(?Tr−1(∇X

?Aξ)

?Ei,

?Ei) = g(

?Tr−1∇X

?ki

?Ei,

?Ei)− g(

?Tr−1

?Aξ∇X

?Ei,

?Ei)

= X(?ki)g(

?Tr−1

?Ei,

?Ei)

= (−1)r−1X(?ki)

?Sir−1

and η(?Tr−1(∇X

?Aξ)ξ) = 0. Hence

tr(?Tr−1 ∇X

?Aξ) = (−1)r−1

n∑i=1

X(?ki)

?Sir−1 = (−1)r−1X(

?Sr),

which completes the proof.

Now, we get the following.

Proposition 2. Let r be an integer such that 1 ≤ r ≤ n. A non-maximal pointp ∈M is r-umbilical if and only if

∀i ∈ 1, . . . , n,?Sir(p) = (r + 1)

?Sr+1(p)?S1(p)

.

Proof. Just observe that?Sr+1 =

?Sir+1 +

?ki

?Sir.

Remark 2. For a large class of null hypersurfaces, namely closed null hypersurfaces,the above proposition cannot be applied globally as they do admit (at least) onemaximal point [14, Remark 10, page 7].

From now on, only Lorentzian ambient manifolds will be in consideration. Re-call that to a normalized null hypersurface (Mn+1, g,N) is associated a (nondegen-erate) metric gη = g+η⊗η [9]. The ambient manifold being Lorentzian, the inducedmetric g on M has signature (0, n). It follows that the hypersurface M equippedwith the associated metric gη is a Riemannian manifold. Let (e0 = ξ, e1, . . . , en)


be a gη-orthonormal basis of Γ(TM) with S (N) = spane1, . . . , en. The diver-

gence of the operator?Tr : Γ(TM)→ Γ(TM) is the vector field div∇(

?Tr) ∈ Γ(TM)

defined as the trace of the End(TM)-valued operator ∇?Tr and given by

div∇(?Tr) = tr(∇

?Tr) =

n∑α,β=0

gα,βη (∇?Tr)(eα, eβ) =

n∑α=0

(∇eα?Tr)eα.

By using the definition of the covariant derivative of a tensor and using (6),

g((∇eα?Aξ)

?Tr−1eα, X) = g(

?Tr−1eα, (∇eα

?Aξ)X)− η(X)BN (eα,

?Aξ

?Tr−1eα).

Hence

n∑α=0

g((∇eα?Aξ)

?Tr−1eα, X) =

n∑α=0

g(?Tr−1eα, (∇eα

?Aξ)X)

− η(X) tr( ?A2ξ

?Tr−1

).

(11)

Proposition 3. For all X ∈ Γ(TM),

g(div?Tr, X) =

r−1∑a=0

n∑i=1

g(R(ei, ξ)

?Taei,

?Ar−1−aξ X

)+

r−1∑a=0

(τN (

?Ar−1−aξ X) tr(

?Aξ

?Ta)− τN (P (

?Aξ

?TaX))

)+ (−1)rη(X)

(n∑i=1

?Sir−1

?k2i − ξ(

?Sr)

) (12)

Proof. Using iterative formula,

div∇(?Tr) = (−1)r div(

?SrI) + div(

?Aξ

?Tr−1)

= (−1)rn∑α=0

((eα ·

?Sr)eα + (∇eα

?Aξ)

?Tr−1eα

)+

?Aξ(div

?Tr−1).

Hence by using (11) we get

g(div∇(?Tr), X) = g(div

?Tr−1,

?AξX) + (−1)rPX(

?Sr)− η(X) tr

( ?A2ξ

?Tr−1

)+

n∑α=0

g( ?Tr−1eα, (∇eα

?Aξ)X

). (13)

By using the Gauss-Codazzi equation (8) with the substitutions

X ←− eα, Y ←− X, Z ←−?Tr−1eα,


we get

g( ?Tr−1eα, (∇eα

?Aξ)X

)= g(R(eα, X)

?Tr−1eα, ξ

)+ g( ?Tr−1eα, (∇X

?Aξ)eα

)+BN (eα,

?Tr−1eα)τN (X)

−BN (X,?Tr−1eα)τN (eα).

(14)

Observe thatn∑α=0

g( ?Tr−1eα, (∇X

?Aξ)eα

)= tr(

?Tr−1 ∇X

?Aξ), (15)

and using this along with (13), (14), (15) and Proposition 1, we obtain

g(div∇(?Tr), X) = g(div

?Tr−1,

?AξX) + (−1)r−1η(X)ξ(

?Sr)

+

n∑α=0

(g(R(eα, X)

?Tr−1eα, ξ)−BN (X,

?Tr−1eα)τN (eα)

)+ τN (X) tr

( ?Aξ

?Tr−1

)− η(X) tr

( ?A2ξ

?Tr−1

)= g(div

?Tr−1,

?AξX) + (−1)r−1η(X)ξ(

?Sr)

+

n∑α=0

g(R(eα, X)?Tr−1eα, ξ)

− τN(P (

?Aξ

?Tr−1X)

)+ τN (X) tr

( ?Aξ

?Tr−1

)− η(X) tr

( ?A2ξ

?Tr−1

).

By using the above iterative formula and Proposition 1, we deduce (12).

Remark 3. Taking r = 1 in (12) and X = ξ, we get

Ric(ξ) = ξ(?S1) + τN (ξ)

?S1 −

n∑i=1

?k2i . (16)

In case the ambient manifold M is a space form and τN = 0, the vector field div?Tr

is TM⊥-valued, that is g(div∇(?Tr), X) = 0 for all X ∈ TM , and

ξ(?Sr) = (−1)r−1 tr

( ?A2ξ

?Tr−1

).

Also (setting X = ξ) the following partial differential equation holds for eachr = 1, . . . , n+ 1

(−1)r−1ξ(?Sr) + τN (ξ) tr(

?Aξ

?Tr−1)− tr(

?A2ξ

?Tr−1) = 0; (17)

or equivalently

ξ(?Sr) + r

?Srτ

N (ξ)−n∑i=1

?k2i

?Sαr−1 = 0.


From the above equation, we recover the well-known fact that for totally umbilicalnull hypersurfaces with principal curvature (umbilicity factor) ρ in a space form,the following partial differential equation holds [12, p. 108]:

ξ(ρ) + ρτN (ξ)− ρ2 = 0.

We also derive the following.

Theorem 1. Let (Mn+1, g,N) be a normalized null hypersurface of a Lorentzianspace form (M(c)n+2, g) with rigged vector field ξ and τN = 0. Then

(a) For each r ∈ 1, . . . , n, M is r-maximal if and only if the endomorphism?A2ξ

?Tr−1 is trace-free.

(b) M is maximal if and only if M is totally geodesic.

(c) If M is r-maximal for some r = 1, . . . , n, then M is s-maximal for all s ≥ r.

Proof. From (17),

(−1)r−1ξ(?Sr)− tr(

?A2ξ

?Tr−1) = 0,

as τN = 0. Then the first item is immediate. Now, take r = 1 in the same equation

(17) to get (b). Finally, if M is r-maximal then by the first item, tr( ?A2ξ

?Tr−1

)= 0.

Hence, Proposition 1 leads to

?S1

?Sr + (r + 1)

?Sr+1 = 0,

which shows that?Sr = 0 implies

?Sr+1 = 0 and the proof is complete.

Recall that a pseudo-Riemannian manifold satisfies the null (resp. the reversenull) convergence condition if Ric(V ) ≥ 0 (resp. Ric(V ) ≤ 0) for any null vectorfield V .

Theorem 2. Let (M, g) be a Lorentzian manifold. If for M the null convergencecondition holds, then for any null hypersurface M of M , M is maximal if and onlyif M is totally geodesic.

Proof. Assume M is maximal. From (16) we have

Ric(ξ) = −n∑i=1

?k2i ≥ 0 as

?S1 = 0 .

Hence each?ki vanishes and M is totally geodesic. The converse is immediate.


3.2 Newton transformations and change of riggingAs stated above, N being a null rigging for M , a vector field N ∈ Γ(TM) is a nullrigging for M if and only if it is defined in an open set containing M and thereexist a smooth function φ on M and a section ζ of TM such that N i = φN + ζwith the properties that φ is nowhere vanishing, being i the inclusion map, and2φη(ζ) + ‖ζ‖2 = 0 along M (see [7] for details on changes in normalizations). Foreach i, set

?

Ei = P?Ei =

?Ei −

1

φg(ζ,

?Ei)ξ and

?

E0 = ξ :=1

φξ.

Lemma 3. (?

E0, . . . ,?

En) is a quasi-orthonormal basis of Γ(TM) which diagonalizes?Aξ with eigenfunctions

?

kα = 1φ

?kα.

Proof. g(?

E0,?

Eα) = 1φg(ξ,

?

Eα) = 0 and g(?

Ei,?

Ej) = g(?Ei,

?Ej) = δij ,

?Aξ ξ = 0 and

?Aξ

?

Ei =?Aξ

?

Ei −1

φ2BN (ζ,

?

Ei)ξ

=?Aξ

?Ei −

1

φ2g(ζ,

?Aξ

?Ei)ξ

=1

φ

?ki

(?Ei −

1

φg(ζ,

?Ei)ξ

)=

1

φ

?ki

?

Ei.

Hence, through the change N = φN + ζ,

?

kα =1

φ

?kα,

?

Hr =1

φr?Hr,

?

Sr =1

φr?Sr,

?

Sir =1

φr?Sir (18)

and we have the next lemma.

Lemma 4. Let (Mn+1, g,N) be a normalized null hypersurface of a Lorentzianmanifold (Mn+2, g). Consider the change of normalization N = φN + ζ. Then

?

Tr =1

φr?Tr −

1

φr+1

r−1∑a=0

(−1)a?Sag

(ζ,

?Ar−aξ

)ξ.

Proof. By use of second relation in (10) we have

?

Tr =

r∑a=0

(−1)a?

Sa?Ar−aξ

= (−1)r?

SrI +

r−1∑a=0

(−1)a?

Sa

(1

φAξ −

1

φ2BN (ζ, ·)ξ

)r−a.


As r − a ≥ 1 and?Aξξ = 0,(

1

φAξ −

1

φ2BN (ζ, ·)ξ

)r−a=

1

φr−aAr−aξ − 1

φr+1−aBN (ζ,

?Ar−a−1ξ )ξ.

This completes the proof.

For each i, in view of (9) we get

∇ ?

Ei

?

Ei = ∇ ?

Ei

?

Ei −1

φBN (

?

Ei,?

Ei)ζ

= ∇ ?Ei

?Ei −

?Ei

(1

φg(ζ,

?Ei)

)ξ − 1

φg(ζ,

?Ei)∇ ?

Eiξ − 1

φg(ζ,

?Ei)∇ξ

?Ei

+1

φg(ζ,

?Ei)∇ξ

1

φg(ζ,

?Ei)ξ −

1

φBN (

?Ei,

?Ei)ζ

= ∇ ?Ei

?Ei −

?kiφg(ζ,

?Ei)

?Ei −

1

φg(ζ,

?Ei)P∇ξ

?Ei −

?kiφPζ

+

[1

φg(ζ,

?Ei)τ

N (?Ei)−

?Ei

(1

φg(ζ,

?Ei)

)+

1

2ξ

(1

φ2g(ζ,

?Ei)

2

)− 1

φg(ζ,

?Ei)η(∇ξ

?Ei)

]ξ

Hence

∇ ?

Ei

?

Ei =1

φ

( ?kig(ζ,

?Ei)

?Ei − g(ζ,

?Ei)

n∑j=1

g(∇ξ?Ei,

?Ej)

?Ej −

?ki

n∑j=1

g(ζ,?Ej)

?Ej

)+∇ ?

Ei

?Ei + η

(∇ ?

Ei

?

Ei −∇ ?Ei

?Ei)ξ, (19)

and

η(∇ ?

Ei

?

Ei −∇ ?Ei

?Ei) =

1

φg(ζ,

?Ei)τ

N (?Ei)−

?Ei

(1

φg(ζ,

?Ei)

)− 1

φg(ζ,

?Ei)η(∇ξ

?Ei) +

1

2ξ

(1

φ2g(ζ,

?Ei)

2

).

Lemma 5. Let (Mn+1, g,N) be a normalized null hypersurface of a Lorentzian

manifold (Mn+2, g) such that for a fixed r, ξ ·?Sir = 0 for i = 1, . . . , n. Consider the

change of normalization N = φN + ζ, ζ|M ∈ Γ(TM), φ ∈ R. Then

div∇(?

Tr) =1

φrdiv∇(

?Tr) + η(div∇(

?

Tr)−1

φrdiv∇(

?Tr))ξ

+(−1)r

φr+1

n∑j=1

n∑i=1

(?Sjr −

?Sir)

(g(∇ξ

?Ei,

?Ej)g(ζ,

?Ei) +

?kig(ζ,

?Ej)) ?Ej .

(20)


In particular for r = 0, . . . , n, div∇( ?Tr)− 1

φr div∇(?Tr) is TM⊥-valued if and only

if for each j = 1, . . . , n,

n∑i=1

(?Sjr −

?Sir)

(g(∇ξ

?Ei,

?Ej)g(ζ,

?Ei) +

?kig(ζ,

?Ej))

= 0. (21)

Proof. Observe that (∇ξ?

Tr)ξ = (−1)r ξ(Sr)ξ. Then

div∇(?

Tr) =

n∑i=1

(∇ ?

Ei

?

Tr)?

Ei

=

n∑i=1

(∇ ?

Ei

?

Tr

?

Ei −?

Tr∇ ?

Ei

?

Ei) + (−1)r ξ(Sr)ξ.

By the second item in Proposition 1,

∇ ?

Ei

?

Tr

?

Ei = (−1)r?

Sir∇ ?

Ei

?

Ei + (−1)r(∇ ?

Ei

?

Sir) ?Ei.

Then thanks to (19) and by direct calculation, we get

∇ ?

Ei

?

Tr

?

Ei =(−1)r

φr+1

?Sir

( ?kig(ζ,

?Ei)

?Ei −

n∑j=1

(g(ζ,

?Ei)g(∇ξ

?Ei,

?Ej) +

?kig(ζ,

?Ej)) ?Ej

)+

1

φr∇ ?Ei

?Tr

?Ei + η(∇ ?

Ei

?

Tr

?

Ei −1

φr∇ ?Ei

?Tr

?Ei)ξ

+ (−1)r( ?Sir

?Ei(1/φ

r)− 1

φg(ζ,

?Ei)ξ(

?Sir/φ

r)) ?Ei

in which the last term vanishes due to ξ ·?Sir = 0 and φ ∈ R. Now (19) and Lemma 4

yield

?

Tr∇ ?

Ei

?

Ei =1

φr?Tr∇ ?

Ei

?

Ei −1

φr+1

r−1∑a=0

(−1)a?Sag(

?Ar−aξ , ∇ ?

Ei

?

Ei)ξ

=(−1)r

φr+1

?ki

?Sirg(ζ,

?Ei)

?Ei

+(−1)r+1

φr+1

n∑j=1

?Sjr(g(ζ,

?Ei)g(∇ξ

?Ei,

?Ej) +

?kig(ζ,

?Ej)) ?Ej

+1

φr?Tr∇ ?

Ei

?Ei + η

( ?Tr∇ ?

Ei

?

Ei −1

φr?Tr∇ ?

Ei

?Ei)ξ.

The desired expression follows from direct substitution. The last claim is immediateby cancelling the screen term represented by the last summation in (20).


Theorem 3. Let (Mn+1, g,N) be a normalized null hypersurface of a Lorentzian

manifold (Mn+2, g), and r ∈ 1, . . . , n such that ξ ·?Sir = 0 for i = 1, . . . , n. Then

div∇( ?Tr)− 1φr div∇(

?Tr) is TM⊥-valued for any change of normalization N = φN+ζ

with φ ∈ R, if and only if any point of M is r-umbilical or both maximal and(r + 1)-maximal.

Proof. Let r ∈ 1, . . . , n and div∇( ?Tr)− 1

φr div∇(?Tr) ∈ Γ(RadTM) for any

change of normalization N = φN + ζ. Then by (21),

n∑i=1

(?Sjr −

?Sir)

(g(∇ξ

?Ei,

?Ej)g(ζ,

?Ei) +

?kig(ζ,

?Ej))

= 0, ∀j = 1, . . . , n.

Consider the particular changes N = N +?El for l = 1, . . . , n. Then for each l,

(?

Sjr −?Slr)g(∇ξ

?El,

?Ej) +

n∑i=1

(?Sjr −

?Sir)

?kiδlj = 0, ∀j = 1, . . . , n,

and setting j = l yields?S1

?Slr − (r + 1)

?Sr+1 = 0.

By Proposition 2 we deduce that any non-maximal point of M is r-umbilical.The converse is straightforward.

3.3 Minkowski integral formulas

Using Newton transformations with respect to the shape operator?Aξ we intro-

duce some Minkowski-type integral formulas on null hypersurfaces of Lorentzianmanifolds carrying some conformal Killing vector field.

Recall that when a manifold M is provided with a linear connection D and Xis a section of the tangent bundle of M , the map DX : Γ(TM)→ Γ(TM) given byTpM 3 Yp 7→ DYpXp is an endomorphism at each point p ∈ M . The divergenceof X (with respect to D) is defined as the trace of DX, that is

divD(X) = tr(DX).

In particular on semi-Riemannian manifolds the default (natural) connection usedin calculating the divergence is the Levi-Civita connection.

Let (Mn+1, g,N) be a normalized null hypersurface of a Lorentzian manifold(Mn+2, g) with rigged vector field ξ and τN = 0. Let∇ denote the linear connectioninduced by the rigging N and assume K ∈ Γ(TM) is a conformal Killing vectorfield with smooth conformal factor 2Φ. For each r ∈ 0, . . . , n+ 1 we have

div∇(?TrK) = tr(∇

?TrK) =

n∑i=1

g(∇ ?Ei

?TrK,

?Ei) + g(∇ξ

?TrK,N).


But

g(∇ ?Ei

?TrK,

?Ei) =

?Ei · g(

?TrK,

?Ei)− g(K,

?Tr∇ ?

Ei

?Ei)− η(

?TrK)BN (

?Ei,

?Ei)

=?Ei · g(K,

?Tr

?Ei) + g(K, (∇ ?

Ei

?Tr)

?Ei)− g(K,∇ ?

Ei

?Tr

?Ei)

− η(?TrK)BN (

?Ei,

?Ei)

= g(K, (∇ ?Ei

?Tr)

?Ei) + (−1)r

?Sirg(∇ ?

EiK,

?Ei)

+ η(K)BN (?Ei,

?Tr

?Ei)− η(

?TrK)BN (

?Ei,

?Ei).

As LKg = 2ϕg we have g(∇ ?EiK,

?Ei) = ϕg(

?Ei,

?Ei). Hence

div∇(?TrK) = g(div∇(

?Tr),K) + ϕ

((−1)r−1

?Sr + tr(

?Tr))

+ η(K) tr( ?Aξ

?Tr − (−1)r

?Sr

?Aξ

)+ η(∇ξ

?TrK)

= g(div∇(?Tr),K) + (−1)r(n− r)

?Srϕ

+ η(K) tr( ?A2ξ

?Tr−1

)+ η(∇ξ

?TrK).

Now using Proposition 1 leads to

div∇(?TrK) = g(div∇(

?Tr),K) + η(∇ξ

?TrK)

+ (−1)r(cr

?Hrϕ+ c′r

?Hr+1η(K)− c′′r

?H1

?Hrη(K)

).

(22)

where

cr = (n− r)(n+ 1

r

), c′r = (n+ 1)

(n

r

), c′′r = (n+ 1)

(n+ 1

r

).

Also, a straightforward computation gives

η(∇ξ?TrK) = (−1)rξ(

?Srη(K)) + (−1)r

?Srτ

N (ξ)η(K)− g(?TrK,ANξ).

We deduce the following.

Theorem 4. Let (Mn+1, g,N) be a normalized null hypersurface of a space-time(Mn+2, g) with rigged vector field ξ and τN = 0, carrying a compactly supportedconformal Killing vector field K with smooth conformal factor 2Φ. Then, for eachr = 1, . . . , n+ 1, the following holds∫

M

(g(div

?Tr−1,K) + η(∇ξ

?Tr−1K)

)dV

= (−1)r∫M

(cr−1

?Hr−1ϕ+ η(K)(c′r−1

?Hr − c′′r−1

?H1

?Hr−1)

)dV, (23)

where dV = iNdV and dV is the (fixed) volume element on M with respect to gand the given orientation.


In particular for horizontal conformal Killing vector fields K on M we have∫M

(g(div

?Tr−1,K)

)dV = (−1)r

∫M

(cr−1

?Hr−1ϕ

)dV. (24)

Proof. Since K is compactly supported, by Stoke’s Theorem,∫M

div∇(?TrK) dV = 0

and (23) is straightforward from (22). Now, assume K to be tangent to the screenstructure S (N). Then η(K) = 0. Also, as τN = 0 we have from Lemma 2 and (4)

that CN (ξ,?Tr−1K) = 0. Therefore

∇ξ(?Tr−1K) =

?∇ξ(

?Tr−1K) + CN (ξ,

?Tr−1K)ξ =

?∇ξ

?Tr−1K ∈ S (N).

Hence η(∇ξ

?Tr−1K

)= 0 and the relation (24) follows (23).

Remark 4. In Theorem 4 and below, the condition compactly supported may beremoved and replaced by compact null hypersurface without boundary.

Corollary 1. Let (Mn+1, g,N) be a normalized null hypersurface of a space-time(Mn+2, g) with rigged vector field ξ and τN = 0, carrying a compactly supportedconformal Killing vector field K with smooth conformal factor 2Φ. Suppose thatfor some r = 1, . . . , n+ 1 the following condition holds∫

M

g(div?Tr−1,K) dV = 0.

Then∫M

(cr−1

?Hr−1Φ + c′r−1g(K,N)

?Hr − c′′r−1g(K,N)

?H1

?Hr−1

)dV

= (−1)r∫M

η(∇ξ?Tr−1K) dV. (25)

In particular, (25) always holds when the ambient space-time (Mn+2, g) hasconstant sectional curvature and for the conformal factor 2Φ we have∫

M

Φ dV = − 1

n

∫M

η(∇ξK) dV. (26)

Moreover, if the conformal Killing vector field K is horizontal then∫M

Φ dV = 0.

Formula (25) is the r-th Minkowski-type formula of the null hypersurface M , with

respect to the shape operator?Aξ.


Proof. Setting∫Mg(div

?Tr−1,K) dV to 0 in (23) leads to (25). From Remark 3 we

know that when the ambient manifold has constant sectional curvature, div∇(?Tr−1)

is TM⊥-valued and the vanishing condition is fulfilled. Finally, set r = 1 in (25)

to get (26), using the fact that c0 = n, c′0 = c′′0 and?H0 = 1. If in addition the

conformal Killing vector field is tangent to the screen structure then by the screenGauss formula and and τN = 0 we have CN (ξ, ·) = 0 and then ∇ξK ∈ S (N), thatis η(∇ξK) = 0 and the last claim follows.

Corollary 2. Let (Mn+1, g,N) be a normalized null hypersurface of a space-time(Mn+2, g) with rigged vector field ξ and τN = 0, carrying a compactly supportedconformal Killing vector field K with smooth conformal factor 2Φ. If M is r-totally

umbilical for some r = 1, . . . , (n + 1) and satisfies both ξ ·?Sir = 0 for i = 1, . . . , n

and the r-th Minkowski-type formula (25), then the same is true for all rigging ofthe form N = ψN + ζ with constant ψ.

Proof. Consider from N a rigging N = ψN+ζ. We pointed out in Theorem 3 that

div∇?

Tr−1 −1

ψr−1div∇

?Tr−1 ∈ Γ(RadTM).

Thus, g(div∇?

Tr−1,K) = 1ψr−1 g(div∇

?Tr−1,K). It follows that for constant ψ we

have ∫M

g(div∇?

Tr−1,K) =1

ψr−1

∫M

g(div∇?Tr−1,K),

which shows that integrals from both sides vanish or not, simultaneously.

4 Physical modelsAs usual, stationary and axisymmetric perfect fluid metrics are studied under theassumption of the existence of a conformal Killing vector field. Let (M4, g) be theEinstein static fluid space-time with metric

ds2 = −dt2 + (1− %2)−1d%2 + %2(dθ2 + sin2 θ dφ2

),

with the fluid four-velocity vector ua = δa0 (a = 0, 1, 2, 3). This space-time admitsa conformal Killing vector field

Ka = (1− %2)1/2 cos t δa0 − %(1− %2)1/2 sin t δa1 .

In fact in this space-time, the relation

% = cos t, t ∈]0,π

2

[defines a compact null hypersurface M for which the kernel of the degenerateinduced metric g is spanned by the null conformal Killing vector field K. In other


words, (M, g) is a compact totally umbilical null hypersurface. Indeed, considerthe vector field

Na = − 1

2%2(1− %2)−1/2

[cos t δa0 + % sin t δa1

].

This is a null rigging with associated screen structure S (N) = span(∂θ, ∂φ). It

is easy to check that τN = 0 and?AK =

[(1 − %2)1/2 sin t

]P where P denotes the

projection morphism of TM onto S (N). The Newton transformations are givenby

?T0 = I,

?Tr = (1− %2)r/2 sinr t

[ r∑a=0

(−1)a(

3

a

)]P, r = 1, 2, 3.

The scale factor is given by Φ = −(1− %2)1/2 sin t. Also, for all r ≥ 1,∫M

g(div?Tr−1,K) dV = 0

and g(K,N) = 1 and by direct calculation, we get∫M

Φ dV = −2π which is non-zero. Observe that the conformal Killing vector field K is not compactly supportedin M .

In general, when interested by perfect-fluid solutions of Einstein’s field equa-tions, it is well-known that there exist coordinates t, x, y, z such that U = ∂y andT = ∂z are two Killing vector fields and in which the metric takes the form

ds2 =1

S2(t, x)

[−dt2 + dx2 + F (t, x)

(P−1(t, x) dy2 + P (t, x)(dz +W (t, x) dy)2

)].

Let us consider the 1-forms θa such that

θ0 =1

S(t, x)dt θ1 =

1

S(t, x)dx

θ2 =1

S(t, x)

√F (t, x)

P (t, x)dy θ3 =

1

S(t, x)

√F (t, x)P (t, x)(dz +W (t, x) dy),

and let Sαβ stand for the components of the Einstein tensor in the θa cobasis.Then the Einstein field equations can be written in terms of the Sαβ and dueto the symmetries inherent to this setting we are led to three inequivalent Liealgebras [16]. The Lie algebra A is given by

[U, T ] = 0, [U,K] =1

2(c+ b)U, [T,K] =

1

2(c− b)T,

where b and c are arbitrary (possibly vanishing) constants and K is a conformalKilling vector field given by

K = ∂t +1

2(c+ b)yU +

1

2(c− b)zT.


The line element in this case has the form

ds2 =1

S2(t, x)

[−dt2 + dx2 + F (x)P−1(x)e−(b+c)t dy2

+ F (x)P (x)e(b−c)t(dz +W (x)e−bt dy)2]

.

Similarly, for the Lie Algebra B we have

[U, T ] = 0, [U ;K] =1

2cU + aT, [T,K] =

1

2cT,

where a is a non-vanishing constant and K is a conformal Killing vector field givenby

K = ∂t +1

2cyU +

(ay +

1

2cz)T.

The corresponding line element has the form

ds2 =1

S2(t, x)

[−dt2 + dx2

+ F (x)e−ct(P−1(x) dy2 + P (x)(dz + [W (x) + at] dy)2

)].

Finally for the Lie algebra VII (so named because it corresponds to the Bianchitype VII in Bianchi’s classification of three-dimensional Lie algebra) the productis defined by

[U, T ] = 0, [U ;K] =1

2cU − aT, [T,K] = aU +

1

2cT,

where a 6= 0 and c are constant and K is a conformal Killing vector field given by

K = ∂t +

(1

2cy + az

)U +

(−ay +

1

2cz

)T.

For each conformal Killing vector field K in above three (non equivalent) Lie alge-bras, the scale factor Φ is given by

Φ = −S,tS.

Now, for each Lie algebra, consider the two distributions

DU,K = spanU,K, DT,K = spanT,K

involving the conformal Killing vector field K. For the Lie algebra A the two dis-tributions DU,K and DT,K are both integrable. For the Lie algebra B, only DT,K

is integrable and for the Lie algebra VII, none of them is integrable. Let M be anycompact null hypersurface without boundary in the perfect-fluid space-time. As-sume N is a rigging for M with screen structure S (N) = DU,K or DT,K accordingto the Lie algebra A or S (N) = DT,K when dealing with the Lie algebra B. ThenK is a horizontal conformal Killing vector field in the rigged null hypersurface M .If the ds2 have constant sectional curvature and τN is vanishing, we get thanks toCorollary 1 that ∫

M

S,tS

dV = 0.


5 Newton transformation of the null hypersurface with respectto the shape operator AN

Throughout this section the normalization is assumed to be closed. In this caseLemma 1 asserts that⟨

ANX,Y⟩−⟨ANY,X

⟩= τN (X)η(Y )− τN (Y )η(X) (27)

for all X,Y ∈ Γ(TM). It follows that the operator AN is symmetric when re-stricted to the screen structure S (N). The ambient manifold will also consid-ered to be Lorentzian which implies that the screen structure is Riemannian. Let(E0 = ξ, E1, . . . , En) be a quasi-orthonormal frame field of TM with S (N) =spanE1, . . . , En. Then the matrix of AN has the form

0 0 · · · 0? k1 · · · 0...

.... . .

...? 0 · · · kn

(28)

where k0, k1, . . . , kn are the principal curvatures of the null hypersurface M withrespect to the shape operator AN . The scalar function H1 = 1

n+1 tr(AN ) is themean curvature of the null hypersurface with respect to AN . For 0 ≤ r ≤ n+1, ther-th mean curvature of the null hypersurface with respect to the shape operator ANis defined by

Hr =

(n+ 1

r

)−1

σr(k0, . . . , kn) and H0 = 1,

and Sr =(n+1r

)Hr.

The characteristic polynomial of AN is given by

P (t) = det(AN − tI) =

n+1∑a=0

(−1)a(n+ 1

a

)Hat

n+1−a.

In a similar way as for the operator?Aξ the Newton transformations Tr (0 ≤ r ≤

n+ 1) of the null hypersurface M with respect to AN are given by

Tr =

r∑a=0

(−1)a(n+ 1

a

)HaA

r−aN .

Inductively,

T0 = I and Tr = (−1)r(n+ 1

r

)HrI +AN Tr−1,

and the following items are straightforward.

Proposition 4. (a) The transformations Tr (0 ≤ r ≤ n + 1) are self-adjointon S (N) and commute with AN .


(b) TrEi = (−1)rSirEi.

(c) tr(Tr) = (−1)r(n+ 1− r)Sr.

(d) tr(AN Tr−1) = (−1)r−1rSr.

(e) tr(A2N Tr−1) = (−1)r(S1Sr + (r + 1)Sr+1).

We prove the following.

Proposition 5. For all X ∈ Γ(TM),

g(div∇(Tr), X) = g(div∇(Tr−1), ANX

)− (−1)r

(g(X,N)ξ(Sr)− S0

r−1X(k0))

+ g((∇ξAN )Tr−1ξ,X)

+

n∑i=1

g(R(Ei, X)Tr−1Ei, N) + η(X)BN (Ei, AN Tr−1Ei)

+ g(ANX,Ei)τN (Ei)− kiτN (ξ)

+ Ei(τN (Tr−1Ei)η(X)− τN (X)η(Tr−1Ei)

)− τN (Tr−1Ei)η(∇EiX) + τN (∇EiX)η(Tr−1Ei)

− τN (∇EiTr−1Ei)η(X) + τN (X)η(∇EiTr−1Ei). (29)

Proof. Using iterative formula,

div∇(Tr) = (−1)r div∇(SrI) + div∇(AN Tr−1)

=

n∑α=0

((−1)r(eα · Sr)eα + (∇eαAN )Tr−1eα

)+AN (div∇(Tr−1)).

Hence, using (27),

g(div∇(Tr), X) = (−1)rPX(Sr) + g(div∇(Tr−1), ANX

)− τN (X)η(div∇(Tr−1))

+ τN (div∇(Tr−1))η(X) +

n∑α=0

g((∇eαAN )Tr−1eα, X

). (30)

Also

g((∇EiAN )Tr−1Ei, X

)= g(Tr−1Ei, (∇EiAN )X

)+ (−1)rη(X)kiS

ir−1B

N (Ei, Ei)

+ τN (∇EiX)η(Tr−1Ei)− τN (Tr−1Ei)η(∇EiX)

+ τN (X)η(∇EiTr−1Ei)− τN (∇EiTr−1Ei)η(X)

+ Ei(τN (Tr−1Ei)η(X)− τN (X)η(Tr−1Ei)

). (31)

Apply the Gauss-Codazzi equation (7) with the substitutions

X → Ei, Y → X, Z → Tr−1Ei,


to get

g(Tr−1Ei, (∇EiAN )X

)= g(R(Ei, X)Tr−1Ei, N

)− kiτN (X)

+ g((∇XAN )Ei, Tr−1Ei

)+ g(ANX,Ei).

(32)

Also, we have

n∑i=1

g(Tr−1Ei, (∇XAN )Ei

)= (−1)r−1

n∑i=1

Sir−1X(ki)

= (−1)r−1(X(Sr − S0

r−1X(k0)).

(33)

Now, feeding back (33) into (32) and then the resulting expression into (31) weobtain by substitution in (30) the desired expression (29).

For the rest of the section we assume τN = 0 which is equivalent to saying thatthe starred entries in the matrix of AN (see (28)) are zero, that is ANξ = 0. Then

g(div∇(Tr), X) =

r−1∑a=0

n∑i=1

g(R(Ei, N)TaEi, A

r−1−aN X

)− η(X)

(tr(

?Aξ AN Tr−1) + (n+ 1− r)−1ξ

(tr(Tr)

)).

In particular when the ambient manifold is Lorentzian with constant sectionalcurvature c we have

g(div∇(Tr), X) = η(X)(c tr(Tr−1)+(−1)rcSr−1−tr(

?Aξ AN Tr−1)−(−1)rξ(Sr)

).

(34)Now we state the following

Theorem 5. Let (Mn+1, g,N) be a closed normalization of a null hypersurface ofa Lorentzian space form (M(c)n+2, g) with rigged vector field ξ and τN = 0. Then,for all r = 0, . . . , n+ 1, div∇(Tr) is TM⊥-valued and

ξ(Sr) + c(n+ 1− r)Sr−1 = (−1)r−1 tr(?Aξ AN Tr−1). (35)

Proof. Let X ∈ χ(M). We have

g(div∇(Tr), X) = g(div∇(Tr), PX)

(34)= η(PX)

(c tr(Tr−1) + (−1)rcSr−1

− tr(?Aξ AN Tr−1)− (−1)rξ(Sr)

)= 0

as η(PX) = 0, which shows that div∇(Tr) is TM⊥-valued. It follows from thesame equation (34) setting X := ξ and using the third item in Proposition 4 that(35) holds.


AcknowledgmentsThe authors are very grateful to the referee for the quality of its relevant remarksin the direction of improving this work.

References

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[2] L. J. Alías, J. H. S. de Lira, J. M. Malacarne: Constant higher-order mean curvaturehypersurfaces in Riemannian spaces. Journal of the Institute of Mathematics of Jussieu5 (04) (2006) 527–562.

[3] L. J. Alías, A. Romero, M. Sánchez: Uniqueness of complete spacelike hypersurfaces ofconstant mean curvature in Generalized Robertson-Walker space-times. Gen. Relat.Grav. 27 (1995) 71–84.

[4] L. J. Alías, A. Romero, M. Sánchez: Spacelike hypersurfaces of constant mean curvatureand Calabi-Bernstein type problems. Tohoku Math. J. 49 (1997) 337–345.

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[6] C. Atindogbe: Normalization and prescribed extrinsic scalar curvature on nullhypersurfaces. Journal of Geometry and Physics 60 (2010) 1762–1770.

[7] C. Atindogbe, L. Berard-Bergery: Distinguished normalization on non-minimal nullhypersurfaces. Mathematical Sciences and Applications E-notes 1 (1) (2013) 18–35.

[8] C. Atindogbe, K.L. Duggal: Conformal screen on null hypersurfaces. Int. J. of Pure andApplied Math. 11 (4) (2004) 421–442.

[9] C. Atindogbe, J.-P. Ezin, J. Tossa: Pseudo-inversion of degenerate metrics. Int. J. ofMathematics and Mathematical Sciences(55) (2003) 3479–3501.

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Authors’ address:Institut de Mathématiques et de Sciences Physiques (IMSP), University ofAbomey-Calavi, 01 BP 613 Porto-Novo, Benin

E-mail: atincyr a©gmail.com, atincyr a©imsp-uac.org

Received: 30th October, 2014Accepted for publication: 16th February, 2015Communicated by: Haizhong Li


The gap theorems for some extremal submanifolds ina unit sphere

Xi Guo and Lan Wu

Abstract. Let M be an n-dimensional submanifold in the unit sphere Sn+p,we call M a k-extremal submanifold if it is a critical point of the functional∫Mρ2k dv. In this paper, we can study gap phenomenon for these sub-

manifolds.

1 Introduction and theoremsLet x : Mn → Sn+p(1) be an n-dimensional compact submanifold in a unit sphere,and let

• e1, . . . , en be a local orthonormal frame of tangent vector field on M ,

• en+1, . . . , en+p be a local orthonormal frame of normal vector field on M ,

• ω1, . . . , ωn, ωn+1, . . . , ωn+p be its dual coframe field.

Then the second fundamental form and the mean curvature vector of M are

A =∑i,j,α

hαijωi ⊗ ωj ⊗ eα, H =∑α

Hαeα =1

n

∑i,α

hαiieα. (1)

We can define trace-free linear maps φα : TqM → TqM by

〈φαX,Y 〉 = 〈AαX,Y 〉 − 〈X,Y 〉〈H, eα〉,

where q ∈M , Aα is the shape operator of eα,

Aα(ei) = −∑j

〈∇eieα, ej〉ej =∑j

hαijej ,

and we define a bilinear map φ : TqM × TqM → TqM⊥ by

2010 MSC: 53C40, 53C24Key words: Extremal functional, Mean curvature, Totally umbilical

86 Xi Guo and Lan Wu

φ(X,Y ) =

n+p∑α=n+1

〈φαX,Y 〉eα. (2)

It’s easy to check that |φ|2 = |A|2 − nH2, where H2 = |H|2 =∑α(Hα)2, and we

denote ρ = |φ|. For any fixed number k with k ≥ 1, we can define the followingfunctional

Fk(x) =

∫M

ρ2k dv. (3)

When k = n2 , it is the Willmore functional. We say x : M → Sn+p is a k-extremal

submanifold if it is a critical point of the functional Fk(x).It seems very interesting to study the gap phenomenon for submanifolds, and

there are some results about compact minimal submanifolds in Sn+p(1), such as in[7]. For Willmore submanifolds, H. Li proved:

Theorem 1. [6] Let M be an n-dimensional compact Willmore submanifold in Sn+p,then ∫

M

[n−

(2− 1

p

)ρ2

]ρn dv ≤ 0. (4)

In particular, if ρ2 ≤ n2−1/p , then either ρ = 0 and M is a totally umbilical sub-

manifold, or ρ2 = n2−1/p . In the latter case, either p = 1 and M is a Willmore torus

Wm,n−m = Sm(√

n−mn

)× Sn−m

(√mn

); or n = 2, p = 2 and M is the Veronese

surface.

And for k-extremal submanifolds, Z. Guo and H. Li, the second author proved:

Theorem 2. [1], [9] Let M be an n-dimensional compact k-extremal submanifoldin Sn+p, 1 ≤ k < n

2 , then∫M

[n−

(2− 1

p

)ρ2

]ρ2k dv ≤ 0. (5)

In particular, if ρ2 ≤ n2−1/p , then either ρ = 0 and M is a totally umbilical subman-

ifold, or ρ2 = n2−1/p . In the latter case, either p = 1, n = 2m and M is a Clifford

torus Cm,m = Sm(√

12

)× Sm

(√12

); or n = 2, p = 2 and M is the Veronese

surface.

In 2011, H. Xu and D. Yang proved the following pinching theorem for sub-manifold which is a critical point of the functional F1(x).

Theorem 3. [8] LetM be an n-dimensional compact 1-extremal submanifold in Sn+p,then there exists an explicit positive constant An depending only on n such that if(∫

M

ρn dv) 2n

< An, (6)

The gap theorems for some extremal submanifolds in a unit sphere 87

An =

min

n(n− 2)2

4n(n− 1)2 + (n− 2)2,

(n− 2)2(n2 − n)

4(n2 − n)(n− 1)2 + (n− 2)2

C(n)−2 (p = 1);

2

3min

n(n− 2)2

4n(n− 1)2 + (n− 2)2,

(n− 2)2(n2 − n)

4(n2 − n)(n− 1)2 + (n− 2)2

C(n)−2 (p ≥ 2),

then M is a totally umbilical submanifold, where C(n) is a positive constant de-pending on n which satisfies:(∫

M

fnn−1 dv

)n−1n ≤ C(n)

∫M

(|∇f |+ (1 +H2)f

)dv (7)

holds for any f ∈ C1(M).

In this paper, we prove the following theorems for the k-extremal submanifoldwhen 1 ≤ k < n

2 :

Theorem 4. Let M be an n-dimensional compact k-extremal submanifold in Sn+p

(n ≥ 3), 1 ≤ k < n2 , then there exists an explicit positive constant An,k depending

only on n and k such that if (∫M

ρn dv) 2n

< An,k, (8)

where

An,k =

C(n)−2 min

n(n− 2)2(2k − 1)

4n(n− 1)2k2 + (2k − 1)(n− 2)2,

(2k − 1)(n− 2)2(n2

2k − n)

4(n2

2k − n)(n− 1)2k2 + (2k − 1)(n− 2)2

(p = 1);

2

3C(n)−2 min

n(n− 2)2(2k − 1)

4n(n− 1)2k2 + (2k − 1)(n− 2)2,

(2k − 1)(n− 2)2(n2

2k − n)

4(n2

2k − n)(n− 1)2k2 + (2k − 1)(n− 2)2

(p ≥ 2),

then M is a totally umbilical submanifold, where C(n) is the same constant asabove.

Theorem 5. Let M be an n-dimensional (n ≥ 3) compact k-extremal submanifoldwith flat normal bundle in Sn+p, 1 ≤ k < n

2 . If ρ2 ≤ n, then either ρ = 0 andM is a totally umbilical submanifold, or p = 1, n = 2m and M is a Clifford torus

Cm,m = Sm(√

12

)× Sm

(√12

).

Remark 1. If k = n2 , then An,k = 0, so our Theorem 4 is trivial when k = n

2 . Ifk = 1, An,1 = An, our Theorem 4 reduces to Xu-Yang’s Theorem 3.


2 Preliminaries and lemmasWe shall make use of the following convention on the range of indices:

1 ≤ A,B,C ≤ n+ p, 1 ≤ i, j, k ≤ n, n+ 1 ≤ α, β, γ ≤ n+ p.

We choose a local orthonormal frame field e1, . . . , en, en+1, . . . en+p along M ,with eii=1,2,...,n tangent to M and eαα=n+1,n+2,...,n+p normal to M . Let ωAbe the corresponding dual coframe, and ωAB be the connection 1-form on Sn+p.Restricted on M , the curvature tensor, the normal curvature tensor can be given by

dωij −∑k

ωik ∧ ωkj = −1

2

∑k,l

Rijklωk ∧ ωl, (9)

dωαβ −∑γ

ωαγ ∧ ωγα = −1

2

∑k,l

R⊥αβklωk ∧ ωl. (10)

and the mean curvature H =∑αH

αeα, where Hα = 1n

∑i h

αii.

The covariant derivative of the second fundamental form is given by∑k

hαij,kωk = dhαij +∑k

hαkiωkj +∑k

hαkjωki +∑β

hβijωβα, (11)∑l

hαij,klωl = dhαij,k +∑l

hαlj,kωli +∑l

hαij,lωlk +∑l

hαil,kωlj +∑β

hβij,kωβα. (12)

In [9], the second author calculated the Euler-Lagrangian equation of Fk(x):

Lemma 1. [9] If x : M → Rn+p(c) be an n-dimensional submanifold in an(n + p)-dimensional space form Rn+p(c). Then for k ≥ 1, M is an extremal sub-manifold of Fk(x) if and only if for n+ 1 ≤ α ≤ n+ p,

0 = −∆(ρ2k−2)Hα + 2(n− 1)∑i

(ρ2k−2),iHα,i

+∑i,j

(ρ2k−2),ijhαij + (n− 1)ρ2k−2∆⊥Hα

+ ρ2k−2

[ ∑i,j,k,β

hαijhβjkh

βki −

∑i,j,β

Hβhαijhβij −

n

2kρ2Hα

].

(13)

Using the above lemma, we can get that:

Lemma 2. If M is an extremal submanifold of Fk(x), then∫M

ρ2k−2

(∆H2 − 2

∑i,j,α

hαijHα,ij

)dv

= 2

∫M

ρ2k−2|∇⊥H|2 dv + 2

∫M

ρ2k−2F dv, (14)

where ∇⊥ is the normal connection on M , and

F :=∑

i,j,k,α,β

Hαhαijhβjkh

βji −

∑j,k,α,β

HαHβhαjkhβjk −

n

2kρ2H2 .


Proof. Multiplying the equation (13) by Hα and integrating over M we obtain

0 = −∫M

∆(ρ2k−2)H2 dv + 2(n− 1)

∫M

∑i,α

(ρ2k−2),iHα,iH

α dv

+

∫M

∑i,j,α

(ρ2k−2),ijhαijH

α dv + (n− 1)

∫M

∑i,α

ρ2k−2Hα,iiH

α dv

+

∫M

ρ2k−2F dv,

(15)

and integrating by parts, we can get∫M

∑i,α

(ρ2k−2),iHα,iH

α dv = −∫M

∑i

ρ2k−2,ii H2 dv −

∫M

∑i,α

ρ2k−2,i Hα

,iHα dv,

so

2

∫M

∑i,α

(ρ2k−2),iHα,iH

α dv = −∫M

∆ρ2k−2H2 dv = −∫M

ρ2k−2∆H2 dv. (16)

Thus we have the following calculations:∫M

∑i,j,α

(ρ2k−2),ijhαijH

α dv = −∫M

∑i,j,α

(ρ2k−2),ihαij,jH

α dv −∫M

∑i,j,α

(ρ2k−2),ihαijH

α,j dv

= −n∫M

∑i,α

(ρ2k−2),iHα,iH

α dv +

∫M

∑i,j,α

ρ2k−2hαij,iHα,j dv

+

∫M

∑i,j,α

ρ2k−2hαijHα,ji dv

=n

2

∫M

ρ2k−2∆H2 dv + n

∫M

ρ2k−2|∇⊥H|2 dv

+

∫M

∑i,j,α

ρ2k−2hαijHα,ij dv, (17)

∫M

∑i,α

ρ2k−2Hα,iiH

α dv =1

2

∫M

ρ2k−2∆H2 dv −∫M

ρ2k−2|∇⊥H|2 dv. (18)

Then (15) becomes

0 = −1

2

∫M

ρ2k−2∆H2 dv +

∫M

ρ2k−2|∇⊥H|2 dv

+

∫M

∑i,j,α

ρ2k−2hαijHα,ij dv +

∫M

ρ2k−2F dv,(19)

so (14) holds.


We also need the following inequalities:

Lemma 3. [8] Let M be an n-dimensional (n ≥ 3) compact submanifold in theunit sphere Sn+p . Then for any f ∈ C1(M), f ≥ 0, t > 0, f satisfies the followinginequality∫

M

|∇f |2 dv ≥ c1(n, t)

(∫M

f2nn−2 dv

)n−2n

− c2(n, t)

∫M

(1 +H2)f2 dv, (20)

where c1(n, t) = (n−2)2

4C(n)2(1+t)(n−1)2 , c2(n, t) = (n−2)2

4t(n−1)2 .

Lemma 4. [4] Let B1, B2, . . . , Bm be symmetric (n × n)-matrices, Set Sαβ =tr(BαBβ), Sα = Sαα, S =

∑α Sα, then∑

α,β

|BαBβ −BβBα|2 +∑α,β

S2αβ ≤

3

2

(∑α

Sα

)2

, (21)

where |B|2 = trBtB.

3 Proof of the theoremsWe also need a Simons’ type formula, which can be found in [6]:

Lemma 5. If x : M → Sn+m be an n-dimensional submanifold, then

1

2∆ρ2 = |∇A|2 − n2|∇⊥H|2 +

∑i,j,k,α

(hαijhαkk,i),j

+ n∑

α,β,i,j,k

Hβφβijφαjkφ

αki + nρ2 + n2H2ρ2

−∑α,β

σ2αβ −

∑α,β,i,j

(R⊥αβij)2 − 1

2∆(nH2),

(22)

where φ is the trace-free tensor which defined above, σαβ =∑i,j φ

αijφ

βij .

From

0 =

∫M

∆ρ2k dv = 2

∫M

∆ρ2ρ2k−2 dv + 2

∫M

〈∇ρ2,∇ρ2k−2〉dv, (23)

and (22), we get that

1

2

∫M

∆ρ2ρ2k−2 dv =

∫M

|∇A|2ρ2k−2 dv + n

∫M

(∑α,i,j

hαijHα,ij −

1

2∆H2)ρ2k−2 dv

+

∫M

Eρ2k−2 dv, (24)

where

E := n∑

α,β,i,j,k

Hβφβijφαjkφ

αki + nρ2 + n2H2ρ2 −

∑α,β

σ2αβ −

∑α,β,i,j

(R⊥αβij)2 .


Using (14) and (23),

0 =

∫M

(|∇A|2 − n|∇⊥H|2)ρ2k−2 dv

+

∫M

(E − nF )ρ2k−2 dv + (2k − 2)

∫M

|∇ρ|2ρ2k−2 dv,

(25)

from Lemma 2.1 in [8] we know that

|∇A|2 − n|∇⊥H|2 =∑α,i,j,k

(φαij,k)2 ≥ |∇ρ|2. (26)

By a direct computation, we have that

E − nF = nρ2 +n2

2kρ2H2 − n

∑α,β,i,j

HαHβφαijφβij −

∑α,β

σ2αβ −

∑α,β,i,j

(R⊥αβij)2, (27)

for∑α,β,i,j

HαHβφαijφβij =

∑i,j

(∑α

Hαφαij

)2

≤(∑i,j

(∑α

φαij

)2)((∑

α

Hα)2)

= ρ2H2,

(28)then

0 ≥ 2k − 1

k2

∫M

|∇ρk|2 dv

+

∫M

[nρ2 +

(n2

2k− n

)H2ρ2 −

∑α,β

σ2αβ −

∑α,β,i,j

(R⊥αβij

)2]ρ2k−2 dv.

(29)

Proof. (Theorem 4) From Lemma 4,

E − nF ≥ nρ2 + (n2

2k− n)ρ2H2 − ηρ4, (30)

where η = min( 32 , 2−

1p ).

From (25), (26) and (30), we know that the following inequality holds,

2k − 1

k2

∫M

|∇ρk|2 dv +

∫M

[n+

(n2

2k− n

)H2 − ηρ2

]ρ2k dv ≤ 0, (31)

and with Lemma 3 and (31), we can get:

0 ≥ 2k − 1

k2c1(n, t)

(∫M

ρ2nn−2k dv

)n−2n

+

(n− 2k − 1

k2c2(n, t)

)(∫M

ρ2k dv

)+

(n2

2k− n− 2k − 1

k2c2(n, t)

)(∫M

H2ρ2k dv

)− η

∫M

ρ2k+2 dv.

(32)


Using the Hölder’s inequality, we have

0 ≥[

2k − 1

k2c1(n, t)− η

(∫M

ρn dv) 2n

](∫M

ρ2nn−2k dv

)n−2n

+

(n− 2k − 1

k2c2(n, t)

)(∫M

ρ2k dv

)+

[n2

2k− n− 2k − 1

k2c2(n, t)

](∫M

H2ρ2k dv

),

let t = (n−2)2(2k−1)4(n−1)2k2 max ( 2k

n2−2kn ,1n ), then Theorem 4 follows.

Proof. (Theorem 5) If M has normal flat bundle, then (29) become

0 ≥ 2k − 1

k2

∫M

|∇ρk|2 dv

+

∫M

[nρ2 +

(n2

2k− n

)H2ρ2 −

∑α,β

σ2αβ

]ρ2k−2 dv

≥∫M

[nρ2 +

(n2

2k− n

)H2ρ2 − ρ4

]ρ2k−2 dv

≥∫M

(n− ρ2)ρ2k dv. (33)

So if ρ ≤ n, then either ρ = 0 and M is a totally umbilical submanifold, or ρ2 = n,for k < n

2 , from (33), we know that H = 0, with the Theorem 3 in [3], we knowthat M lies in a (n + 1)-dimensional unit sphere, so the Theorem 5 follows fromthe Theorem 2.

AcknowledgementsThe authors thank the referee for useful comments, and they would like to thankProfessor Haizhong Li for his useful advice. The first author was supported byNSFC (Grant No. 11426097).

References

[1] Z. Guo, H. Li: A variational problem for submanifolds in a sphere. Monatsh. Math. 152(2007) 295–302.

[2] D. Hoffman, J. Spruck: Sobolev and isoperimetric inequalities for Riemanniansubmanifolds. Comm. Pure. Appl. Math. 27 (1974) 715–727.

[3] K. Kenmotsu: Some remarks on minimal submanifolds. Tohoku. Math. J. 22 (1970)240–248.

[4] A.-M. Li, J.-M. Li.: An intrinsic rigidity theorem for minimal submanifolds in a sphere.Arch. Math. 58 (1992) 582–594.

[5] H. Li.: Willmore hypersurfaces in a sphere. Asian. J. Math. 5 (2001) 365–378.

[6] H. Li.: Willmore submanifolds in a sphere. Math. Res. Letters 9 (2002) 771–790.


[7] J. Simons.: Minimal varieties in Riemannian manifolds. Ann. of Math. 88 (1968) 62–105.

[8] H.-W. Xu, D. Yang.: The gap phenomenon for extremal submanifolds in a Sphere.Differential Geom and its Applications 29 (2011) 26–34.

[9] L. Wu.: A class of variational problems for submanifolds in a space form. Houston J.Math. 35 (2009) 435–450.

Authors’ addressSchool of Mathematics and statistics, Hubei University, Wuhan 430062,P.R.China, Department of Mathematics, Renmin University of China, Beijing100872, P. R. China

E-mail: guoxi a©hubu.edu.cn, wulan a©ruc.edu.cn

Received: 24th February, 2015Accepted for publication: 10th March, 2015Communicated by: Haizhong Li


Contents of Previous Volumes∗

No. 1, Vol. 18 (2010)Editorial

From the Editor-in-Chief

Research papers

David Saunders: Some geometric aspects of the calculus of variations in severalindependent variables

Yong-Xin Guo, Chang Liu and Shi-Xing Liu: Generalized Birkhoffian realizationof nonholonomic systems

Mike Crampin: Homogeneous systems of higher-order ordinary differentialequations

Survey paper

Olga Krupková: Geometric mechanics on nonholonomic submanifolds

Book review

Geoff Prince: Classical Mechanics: Hamiltonian and Lagrangian Formalism byAlexei Deriglazov

No. 1, Vol. 19 (2011)Research papers

István Pink, Zsolt Rábai: On the diophantine equation x2 + 5k17l = yn

Lorenzo Fatibene, Mauro Francaviglia: General theory of Lie derivatives forLorentz tensors

Martin Swaczyna: Several examples of nonholonomic mechanical systems

Larry M. Bates, James M. Nester: On D’Alembert’s Principle

∗Full texts in pdf are available free on-line at http://cm.osu.cz.

96

Jing Zhang, Bingqing Ma: Gradient estimates for a nonlinear equation∆fu+ cu−α = 0 on complete noncompact manifolds

Book review

Tom Mestdag: Geometry of Nonholonomically Constrained Systemsby R. Cushman, H. Duistermaat and J. Sniatycki

No. 2, Vol. 19 (2011)Geometrical aspects of variational calculus on manifolds

Guest Editor: László Kozma

Editorial

Minicourse

David J. Saunders: Homogeneous variational problems: a minicourse

Research papers

Lorenzo Fatibene, Mauro Francaviglia, Silvio Mercadante: About BoundaryTerms in Higher Order Theories

Zoltán Muzsnay, Péter T. Nagy: Tangent Lie algebras to the holonomy groupof a Finsler manifold

József Szilasi, Anna Tóth: Conformal vector fields on Finsler manifolds

Monika Havelková: A geometric analysis of dynamical systems with singularLagrangians

Wlodzimierz M. Tulczyjew: Variational formulations I: Statics of mechanicalsystems.

Book review

Jaroslav Dittrich: Mathematical results in quantum physics edited by P. Exner

No. 1, Vol. 20 (2012)Guest editor: Marcella Palese

Editorial

Research papers

L. Fatibene, M. Francaviglia, S. Garruto: Do Barbero-Immirzi connections exist indifferent dimensions and signatures?

M. Francaviglia, M. Palese, E. Winterroth: Locally variational invariant fieldequations and global currents: Chern-Simons theories

Monika Havelková: Symmetries of a dynamical system represented by singularLagrangians

Contents of Previous Volumes 97

Zoltán Muzsnay, Péter T. Nagy: Witt algebra and the curvature of the Heisenberggroup

Olga Rossi, Jana Musilová: On the inverse variational problem in nonholonomicmechanics

David J. Saunders: Projective metrizability in Finsler geometry

Conference announcements


Fa-en Wu, Xin-nuan Zhao: A New Variational Characterization Of CompactConformally Flat 4-Manifolds

Florian Luca: On a problem of Bednarek

Hemar Godinho, Diego Marques, Alain Togbe: On the Diophantine equationx2 + 2α5β17γ = yn

Emanuel Lopez, Alberto Molgado, Jose A. Vallejo: The principle of stationaryaction in the calculus of variations

Elisabeth Remm: Associative and Lie deformations of Poisson algebras

Alexandru Oana, Mircea Neagu: Distinguished Riemann-Hamilton geometryin the polymomentum electrodynamics



Thomas Friedrich: Cocalibrated G2-manifolds with Ricci flat characteristicconnection

Junchao Wei: Almost Abelian rings

Bingqing Ma, Guangyue Huang: Eigenvalue relationships between Laplacians ofconstant mean curvature hypersurfaces in Sn+1

Johannes Schleischitz: Diophantine Approximation and special Liouville numbers

Mileva Prvanovic: The conformal change of the metric of an almost Hermitianmanifold applied to the antiholomorphic curvature tensor


Contents of previous Volumes

98


Johanna Pék: Structure equations on generalized Finsler manifolds

Ross M. Adams, Rory Biggs, Claudiu C. Remsing: Control Systems on theOrthogonal Group SO(4)

Tadeusz Pezda: On some issues concerning polynomial cycles

Stephen Bruce Sontz: A Reproducing Kernel and Toeplitz Operators in theQuantum Plane

Serge Preston: Supplementary balance laws for Cattaneo heat propagation

Susil Kumar Jena: Method of infinite ascent applied on −(2p ·A6) +B3 = C2

Erratum



Satish Shukla: Fixed point theorems of G-fuzzy contractions in fuzzy metricspaces endowed with a graph

Xuerong Qi, Linfen Cao, Xingxiao Li: New hyper-Kähler structures on tangentbundles

Benjamin Cahen: Stratonovich-Weyl correspondence for the Jacobi group

Florian Luca, Volker Ziegler: A note on the number of S-Diophantine quadruples

Albo Carlos Cavalheiro: Existence of entropy solutions for degenerate quasilinearelliptic equations in L1

Werner Georg Nowak: Lower bounds for simultaneous Diophantine approximationconstants

Karl Dilcher, Larry Ericksen: Reducibility and irreducibility of Stern(0, 1)-polynomials



Silvestru Sever Dragomir: Bounds for Convex Functions of Čebyšev FunctionalVia Sonin’s Identity with Applications

Mikihito Hirabayashi: A determinant formula for the relative class number of animaginary abelian number field

Benjamin Küster: Discontinuity of the Fuglede-Kadison determinant on a groupvon Neumann algebra

Contents of Previous Volumes 99

Reinhardt Euler, Luis H. Gallardo, Florian Luca: On a binary recurrent sequenceof polynomials

Michal Čech, Jana Musilová: Symmetries and currents in nonholonomic mechanics

Josua Groeger: Super Wilson Loops and Holonomy on Supermanifolds


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Anass Ourraoui: On a class of nonlocal problem involving a critical exponent 47

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