Introduction - cvgmt: homecvgmt.sns.it/media/doc/paper/2660/CiraoloMaggi.pdf · Introduction We investigate the geometry of compact boundaries with almost constant mean curvature

ON THE SHAPE OF COMPACT HYPERSURFACES WITH ALMOSTCONSTANT MEAN CURVATURE

G. CIRAOLO AND F. MAGGI

Abstract. The distance of an almost constant mean curvature boundary from a finite familyof disjoint tangent balls with equal radii is quantitatively controlled in terms of the oscillationof the scalar mean curvature. This result allows one to quantitatively describe the geometry ofvolume-constrained stationary sets in capillarity problems.

1. Introduction

We investigate the geometry of compact boundaries with almost constant mean curvaturein Rn+1, n ! 2. Beyond its intrinsic geometric interest, this problem is motivated by thedescription of equilibrium shapes (volume-constrained stationary sets) of the classical Gaussfree energy used in capillarity theory [Fin86], and consisting of a dominating surface tensionenergy plus a potential energy term. Our analysis leads to new stability estimates describing ina quantitative way the distance of these shapes from compounds of tangent balls of equal radii.

1.1. Main result. Given a connected bounded open set ! " Rn+1 (n ! 2) with C2-boundary,we denote by H the scalar mean curvature of !! with respect to the outer unit normal "! to !(normalized so that H = n if ! = B = {x # Rn+1 : |x| < 1}), and we introduce the Alexandrov’sdeficit of !,

#(!) =$H %H0$C0(!!)

H0, where H0 =

nP (!)

(n+ 1)|!| . (1.1)

This is a scale invariant quantity (i.e. #(!) = #($!) for every $ > 0) with the property that#(!) = 0 if and only if ! is a ball (Alexandrov’s theorem). The motivation for the particularvalue of H0 considered in the definition of #(!) is that if H is constant on !!, then by thedivergence theorem it must be H = H0 (see (2.16) below). Here and in the following, Hk

stands for the k-dimensional Hausdor" measure on Rn+1, |!| is the Lebesgue measure (volume)of !, and P (E) is the distributional perimeter of a set of finite perimeter E " Rn+1 (so thatP (E) = Hn(!E) whenever E is an open set with Lipschitz boundary).

Motivated by applications to geometric variational problems (see section 1.2) we want todescribe the shape of sets ! with small Alexandrov’s deficit. This is a classical question inconvex geometry, where the size of #(!) for ! convex has been related to the Hausdor" distanceof !! from a single sphere in various works, see [Sch90, Arn93, Koh00]. However one shouldkeep in mind that, as soon as convexity is dropped o", the smallness of #(!) does not necessarilyimply proximity to a single ball. Indeed, by slightly perturbing a given number of spheres ofequal radii connected by short catenoidal necks, one can construct open sets {!h}h!N withthe property that, as h & ', #(!h) & 0, while the necks contract to points and the sets !h

converge to an array of tangent balls, see [But11, BM12] (and, more generally for this kind ofconstruction, the seminal papers [Kap90, Kap91]). At the same time, if #(!) is small enough interms of n and the largest principal curvature of !!, then ! must be close to a single ball. Moreprecisely, denoting by A the second fundamental form of !!, in [CV15] it is proved the existenceof c(n, $A$C0(!!)) > 0 such that if #(!) ( c(n, $A$C0(!!)), then the in-radius and out-radius of

1

2 G. CIRAOLO AND F. MAGGI

z2

#!/4(n+1)

#!/4(n+1)

#!

z1

Figure 1. The situation in Theorem 1.1, with # = #(!) and % = 1/2(n+2). The greyregion depicts !#G (whose area is of order #!), while (Id + &G"G)($) is depicted by abold line. The spheres !Bzj ,1 are at a distance of order #!/4(n+1), while $ is obtained

from !G by removing two spherical caps of diameter #!/4(n+1).

! must satisfyrout(!)

rin(!)% 1 ( C(n, $A$C0(!!)) #(!) , (1.2)

where the linear control in terms of #(!) is sharp, as shown for example by taking a sequenceof almost-round ellipsoids. In light of the examples from [But11], an assumption like #(!) (c(n, $A$C0(!!)) is necessary in order to expect ! to be close to a single ball.

Our goal here is to address the situation when a di"erent kind of smallness assumption on#(!) is considered. Indeed, we are just going to assume that #(!) is small with respect to thescale invariant quantity

Q(!) =P (!)n+1

(n+ 1)n+1|!|n |B| =!P (!)

P (B)

"n+1 ! |B||!|

"n.

Notice that by the Euclidean isoperimetric inequality

P (!) ! (n+ 1) |B|1/(n+1) |!|n/(n+1) = P (B)! |!||B|

"n/(n+1), (1.3)

one always has Q(!) ! 1, and that

Q#a union of L disjoint balls of equal radii

$= L , )L # N , L ! 1 .

Hence, one may expect the integer part of Q(!) to indicate the number of balls of radius n/H0

that should be approximating !: and indeed, given L # N, L ! 1, and a # [0, 1), in Theorem1.1 we are going to prove that if Q(!) ( L+ 1% a (so that the normalized perimeter of ! is atad less than the normalized perimeter of (L + 1)-many balls) and #(!) ( #(n,L, a), then ! isclose (in the various ways specified below, and quantitatively in terms of powers of #(!)) to acompound of at most L-many mutually tangent balls of radius n/H0.

Before stating Theorem 1.1 it seems convenient to rescale ! in such a way that the referenceballs have unit radius, that is, we rescale ! (as we can always do) in such a way that

H0 = n and thus P (!) = (n+ 1)|!| , Q(!) =|!||B| =

P (!)

P (B).

Here and in the following we also set Bx,r = {y # Rn+1 : |y % x| < r} (so that B = B0,1) and,given two compact sets K1,K2 in Rn+1, we define their Hausdor" distance as

hd(K1,K2) = max%maxx!K1

dist(x,K2),maxx!K2

dist(x,K1)&.

HYPERSURFACES WITH ALMOST CONSTANT MEAN CURVATURE 3

Moreover, we let

% =1

2(n+ 2), (1.4)

and we refer readers to the beginning of section 2 for our conventions about constants.

Theorem 1.1. Given n,L # N with n ! 2 and L ! 1, and a # (0, 1], there exists a positiveconstant c(n, L, a) > 0 with the following property. If ! is a bounded connected open set withC2-boundary in Rn+1 such that H > 0 and

H0 = n , P (!) ( (L+ 1% a)P (B) , #(!) ( c(n,L, a) ,

then there exists a finite family {Bzj ,1}j!J of mutually disjoint balls with #J ( L such that ifwe set

G ='

j!JBzj ,1 ,

then

|!#G||!| ( C(n)L2 #(!)" , (1.5)

|P (!)%# J P (B)|P (!)

( C(n)L2 #(!)" , (1.6)

maxx!!G dist(x, !!)

diam(!)( C(n)L #(!)" , (1.7)

hd(!!, !G)

diam(!)( C(n)L3/n #(!)"/4n

2(n+1) . (1.8)

Moreover, there exists an open subset $ of !G and a function & : $ & R with the followingproperties. The set !G \$ consists of at most C(n)L-many spherical caps whose diameters arebounded by C(n) #(!)"/4(n+1). The function & is such that (Id + & "G)($) " !! and

$&$C1,!(") ( C(n, ') , )' # (0, 1) , (1.9)

$&$C0(")

diam(!)( C(n)L #(!)" , $*&$C0(") ( C(n)L2/n #(!)"/8n(n+1) , (1.10)

Hn(!! \ (Id + & "G)($))

P (!)( C(n)L4/n #(!)"/4n(n+1) , (1.11)

where (Id + & "G)(x) = x+ &(x) "G(x) and "G is the outer unit normal to G. Finally:

(i) if # J ! 2, then for each j # J there exists ( # J , ( += j, such that

dist(!Bzj ,1, !Bz",1)

diam(!)( C(n) #(!)"/4(n+1) , (1.12)

that is to say, each ball in {Bzj ,1}j!J is close to be tangent to another ball from thefamily;

(ii) if there exists ) # (0, 1) such that

|Bx,r \ !| ! ) |B| rn+1 , )x # !! , r < ) , (1.13)

and #(!) ( c(n,L,)), then #J = 1, that is, ! is close to a single ball.

A first consequence of Theorem 1.1 is that examples of the kind constructed in [But11] areactually the only possible examples of boundaries with almost-constant mean curvature which arenot close to a single sphere. Conversely, the examples of [But11] show that Theorem 1.1 providesa qualitatively optimal information on sets with small Alexandrov’s deficit. But of course, thestrongest aspect of Theorem 1.1 is its quantitative nature. It is precisely this last feature which


is needed in order obtain explicit (although arguably non-sharp) orders of magnitude in thedescription of capillarity droplets, see Proposition 1.2 below and the discussion after it.

A second remark is that, thanks to conclusion (i) and up to a translation of the balls Bzj ,1

of the order of #(!) appearing in (1.12), one can work with a reference configuration G suchthat !G is connected, that is to say, for every j # J one can assert that !Bzj ,1 is tangent to!Bz",1 for some ( += j. Of course, in doing so, the various smallness estimates (1.5)–(1.11) willbe of the same order of #(!) as in (1.12).

The use of the constant a should help to stress the “quantization” e"ect of the perime-ter/energy (and of the volume) that happens under the small deficit assumption. Depending onthe situation, one could be already satisfied of working with the simpler statement correspondingto the choice a = 1.

We also comment on assumption (ii). Our idea here is to provide more robust smallnesscriterions for proximity to a single ball than #(!) ( #(n, $A$C0(!!)). The first criterion justamounts in asking that P (!) ( (2% a)P (B), for a # (0, 1]. The interest of the second criterionis immediately understood if one considers that local minimizers of the capillarity energy satisfyuniform volume density estimates. Of course, a third criterion for proximity to a single ball isrequiring the perimeter upper bound P (!) ( 2% a (which corresponds to taking L = 1).

We now illustrate the proof of Theorem 1.1. Our argument is based on Ros’ proof ofAlexandrov’s theorem [Ros87], which follows closely the ideas of Reilly [Rei77], and is basedon the following Heintze-Karcher inequality [HK78]: if ! is a bounded connected open set withC2-boundary in Rn+1 with H > 0, then

(

!!

n

HdHn ! (n+ 1)|!| . (1.14)

Now, if H is constant, then it must be H = H0 = nP (!)/(n + 1)|!|, so that ! must be anequality case in (1.14). By exploiting Reilly’s identity [Rei77], Ros proves that if equality holdsin (1.14), then the solution f of

)#f = 1 in ! ,

f = 0 on !! ,

satisfies *2f = Id/(n + 1) on ! and |*f | = n/H0(n + 1) on !!. By *2f = Id/(n + 1) on !,there exist x0 # Rn+1 and c < 0 such that f(x) = c+ |x% x0|2/2(n+ 1) for every x # !, i.e. !is the ball of center x0 and radius r =

*%2(n+ 1)c, while, by |*f | = n/H0(n + 1) on !!, it

must be r = n/H0. When H is not constant, one can still infer from the proof of (1.14) that

C(n) |!| #(!)1/2 !(

!

+++*2f % Id

n+ 1

+++ , (1.15)

C(n)! n

H0

"2P (!) #(!) !

(

!!

+++n/H0

n+ 1% |*f |

+++2, (1.16)

where the second estimate holds if #(!) ( 1/2, and where *f = |*f | "! += 0 on !!.The problem of exploiting (1.15) and (1.16) in the description of ! has some analogies with

the quantitative analysis of Serrin’s overdetermined problem [Ser71] addressed in [BNST08]. Inour terminology, the main result from [BNST08] states that, if H0 = n and for some t > 0 onehas (

!!

+++1

n+ 1% |*f |

+++ ( P (!) t , $*f$C0(!!) (1 + t

n+ 1, (1.17)

then there exist finitely many disjoint balls {Bxi,ri}mi=1 such that

+++!#m'

i=1

Bxi,ri

+++(n+1)/2

+ max1"i"m

++ri % 1++ ( C(n, diam(!)) t# , * =

1

4n+ 13. (1.18)


Because of the uniform upper bound on |*f | in (1.17), it is not clear if one can take advantageof this result in the proof of Theorem 1.1. At the same time, we have a di"erent conditionat our disposal, namely (1.15), and by combining (1.15) with a global Lipschitz estimate forf (which is based on [CGS94], and exploits the geometric assumption that H > 0 on !!), weobtain a more precise control than (1.18) on the distance of ! from a finite family of balls.(Indeed, the power % in (1.5) is larger than the power 2*/(n+ 1) appearing in (1.18).) Finally,in Serrin’s overdetermined problem the limiting balls need not to be tangent (sets ! with smallt may contain arbitrarily long connecting necks) and one does not expect Hausdor" estimateslike (1.7) and (1.8) to hold (a set ! with small t may contain small inclusions of large meancurvature). In other words, although (1.18) provides a qualitatively sharp information in thecontext of Serrin’s problem, in the case of Alexandrov’s theorem one expects, and thus wants toobtain, stronger information on !.

Coming back to the proof of Theorem 1.1, the first step consists in proving a qualitativeresult, see Theorem 2.5 below. Indeed, by combining a compactness argument, (1.15) and (1.16)with Reilly’s identity, Pohozaev’s identity, and Allard’s regularity theorem (for integer rectifiablevarifolds with bounded distributional mean curvature) one comes to prove the following fact: if{!h}h!N is a sequence of open, bounded and connected sets with C2-boundary in Rn+1, n ! 2,such that for some L # N, L ! 1,

limh#$

#(!h) = 0 , suph!N

Q(!h) < L+ 1 ,

then, setting

$h =P (!h)

(n+ 1)|!h|, !%

h = $h!h ,

and up to translations, one has

limh#$

hd(!!%h, !G) + |P (!%

h)% P (G)| = 0 ,

where G is the union of at most L-many disjoint balls with unit radii, and with !G connected.Moreover, for every h large enough there exist open sets $h " !G (obtained by removing from!G at most C(n)L-many spherical caps) and functions &h # C1,$($h) for every ' # (0, 1) suchthat (Id + &h "G)($h) " !!%

h, and

limh#$

hd($h, !G) = 0 , $&h$C1,!("h) ( C(n, ') , limh#$

$&h$C1("h) = 0 .

This qualitative stability result, Theorem 2.5, is not needed in the proof of Theorem 1.1, and ofcourse it is actually a corollary of it. We have nevertheless opted for including a direct discussionof it for the following reasons. First of all, it is a result of independent interest and possibleusefulness, so it seems interesting to have a shorter proof of it. Secondly, by having Theorem 2.5at hand one is able to clean up to some later quantitative arguments and obtain better estimates.Thirdly, Theorem 1.1 is actually proved by quantitatively revisiting the proof of Theorem 2.5,and therefore the separate treatment of the latter should makes more accessible the argumentused in proving the former.

In this direction the main di%culty arises in the application of the area excess regularitycriterion of Allard, which is needed to parameterize a large portion of !! over a large portion of!G. A key point here is quantifying the size of Hn(!!,Bx,r) on a range of scales r proportionalto a suitable power of #(!) and at points x # !! su%ciently close to !G. We address this issueby carefully partitioning Rn+1 into suitable polyhedral regions associated to the balls Bzj ,1, andby then performing inside each of these regions a calibration type argument with respect to thecorresponding ball Bzj ,1 (see, in particular, step six of the proof of Theorem 1.1).

Summarizing, Theorem 1.1 is proved by combining a mix of di"erent ideas from elliptic PDEtheory, global geometric identities, and geometric measure theory, and it contains a quantitative(and qualitatively sharp) description of boundaries with almost-constant mean curvature.


1.2. An application to capillarity surfaces. The study of the basic capillarity-type energyfunctional in Rn+1 leads to consider sets ! with small Alexandrov’s deficit. Indeed, given apotential energy density g : Rn+1 & R, in capillarity theory one considers the free energy

F(!) = P (!) +

(

!g(x) dx ,

and its volume-constrained stationary points and local/global minimizers. Capillarity phenom-ena are characterized by the dominance of surface tension over potential energy, which is thecase when the volume parameter m = |!| is small (as surface tension is of order mn/(n+1), whilepotential energy is typically of order m). Under mild assumptions on g (essentially, coercivity atinfinity, g(x) & ' as |x| & '), one can show the existence of volume-constrained global mini-mizers of F of any fixed volume. In particular, if !m is such a global minimizer with |!m| = m,then by comparison with a ball B(m) of volume m one sees that

P (!m) ( P (B(m)) +

(

B(m)\!m

g(x) dx ,

that is, the isoperimetric deficit #iso(!m) of !m is small in terms of m,

#iso(!m) =P (!m)

P (B(m))% 1 ( C(n)

mn/(n+1)

(

B(m)\!m

g(x) dx - C(n, g)m1/(n+1) .

By the quantitative isoperimetric inequality [FMP08, FMP10], one finds xm # Rn+1 such that

! |!m#(xm +B(m))|m

"2( C(n) #iso(!m) ,

so that, in conclusion, !m has to be close (in a normalized L1-sense) to a ball of volume m.This observation is the starting point of the analysis performed in [FM11], where the proximityof !m to a ball of volume m is quantified, under increasingly stronger smoothness assumptionson g, in increasingly stronger ways. For example, if g # C1

loc(Rn+1) and m ( m0(n, g), then !m

is shown to be convex and !!m is proved to a C2,$-small normal deformation of xm + !B(m),with explicit quantitative bounds on the C2,$-norm of this deformation in terms of m.

When dealing with volume-constrained local minimizers or stationary points of F one cannotrely anymore on the quantitative isoperimetric inequality, as one is not given the energy com-parison inequality with B(m). However, in this more general context, Alexandrov’s deficit turnsout to be small in terms of the volume parameter m, thus opening the way for the applicationof Theorem 1.1.

Let us recall that given a vector field X # C$c (Rn+1;Rn+1), and denoted by ft the flow

generated by X, then the first variation of F at ! along X is defined as

#F(!)[X] =d

dt

+++t=0

F(ft(!)) . (1.19)

One says that a set of finite perimeter ! " Rn+1 is a volume-constrained stationary point of F if#F(!)[X] = 0 for every X # C$

c (Rn+1;Rn+1) such that |ft(!)| = |!| for every t small enough.The following proposition, combined with Theorem 1.1, provides a complete description of suchstationary boundaries, and its simple proof is presented in section 3.

Proposition 1.2. Let g # C1loc(Rn+1), R0 > 0, and ! be an open set with C2-boundary such

that ! " BR0. If ! is a volume-constrained stationary point of F with |!| = m, then

#(!) ( C%(n) $g$C1(BR0 )m1/(n+1) , (1.20)

for some constant C%(n).


Under the assumptions of Proposition 1.2, let us now pick L # N, L ! 1, and a # (0, 1],define c(n,L, a) as in Theorem 1.1, and assume that

Q(!) ( L+ 1% a , m (! c(n,L, a)

C%(n) $g$C1(BR0 )

"n+1.

In this way, by Theorem 1.1 and (1.20), there exists a finite family {Bzj ,1}j!J of disjoint ballssuch that, looking for example at (1.5) and setting G =

,j!J Bzj ,1,

|!#G%||!| =

|!%#G||!%| ( C(n)L2 $g$"C1(BR0 )

m"/(n+1) ,

where !% = (H0/n)!, and thus G% =,

j!J Bwj ,n/H0. Notice that this proves a quantization of

the volume of !, in the sense that |!%| is close to #J |B| with an error of order Cm"/(n+1),where C = C(n,L, $g$C1(BR0 )

). Similar results carrying di"erent geometric information areobtained from the other estimates appearing in Theorem 1.1.

In conclusion, the quantitative side of Theorem 1.1 significantly strengthens the purely qual-itative analysis that one could obtain by exploiting compactness arguments only, as it providesexplicit orders of magnitude for the errors one makes in approximating !% with a unit ballscompound.

We finally notice that !% will be close to a single ball as soon as volume-constrained station-arity is strengthened into some local minimality property. For example, it will su%ce to requirethat F(!) ( F(E) whenever |E| = |!| and !E " I%(!!) = {x # Rn+1 : dist(x, !!) < +}, with+ = +0 |!|/P (!) for some +0 > 0. Notice that although E = B(m) is not an admissible competi-tor in this local minimality condition, thus ruling out the possibility of applying [FMP08], ! willnevertheless be a volume-constrained stationary set for F . Moreover, by a standard argumentexploiting the local minimality of !, one obtains volume density estimates for !% which makepossible to apply statement (ii) in Theorem 1.1.

1.3. Organization of the paper. The proof of Theorem 1.1 and of Proposition 1.2 are dis-cussed, respectively, in section 2 and section 3. In Appendix A we discuss the relation of theAlexandrov’s stability problem with the study of almost-umbilical surfaces initiated by De Lellisand Muller in [DLM05].

Acknowledgment: We thank Manuel Ritore for stimulating the writing of Appendix A. Thiswork has been done while GC was visiting the University of Texas at Austin under the supportof NSF-DMS FRG Grant 1361122, of a Oden Fellowship at ICES, the GNAMPA of the IstitutoNazionale di Alta Matematica (INdAM), and the FIRB project 2013 “Geometrical and Qual-itative aspects of PDE”. FM is supported by NSF-DMS Grant 1265910 and NSF-DMS FRGGrant 1361122.

2. Proof of Theorem 1.1

We begin by gathering various assumptions, preliminaries, and conventions.

Constants: The symbol C denotes a generic positive constant whose value is independent fromn and !. We use the symbols C0, C1, etc. for constants whose specific value is referred to inmultiple occasions (see, for instance (2.18) below). We denote by C(n) and c(n) generic positiveconstants whose value does depend on n, but is independent from !, with the idea that C(n)stands for a “large” constant, and c(n) stands for a “small” constant. Similar conventions holdfor C(n,L), etc.

Assumptions on !: Thorough this section we always assume that

! " Rn+1, n ! 2, is a bounded connected open set

with C2-boundary with H > 0 on !! .(2.1)


From a certain point of our argument on we shall assume that (as one can always do up to ascaling)

(n+ 1)|!| = P (!) . (2.2)

Recall that, by the Euclidean isoperimetric inequality (see (1.3)), (2.2) implies

|B| ( |!| , P (B) ( P (!) , (2.3)

where B = {x # Rn : |x| < 1}. Moreover, (2.2) is equivalent to H0 = n, so that

#(!) = n&1 $H % n$C0(!) ,

with #(!) = 0 if and only if ! is a ball of unit radius. Indeed, our convention for the scalarmean curvature H is that

(

!!div !!X dHn =

(

!!(X · "!)H dHn , )X # C1

c (Rn+1;Rn+1) ,

and thus the scalar mean curvature of B is equal to n. In addition to (2.2) we also assume that

P (!) ( (L+ 1% a)P (B) , where L # N, L ! 1, a # (0, 1]. (2.4)

Note that, by combining (2.4) with (2.2) one finds

|!| ( (L+ 1% a)|B| . (2.5)

We shall work under the assumption that #(!) ( c(n,L, a) for a suitably small positive constantc(n,L, a) ( 1/2: in particular,

n

2( H(x) ( 2n , )x # !! . (2.6)

By Topping’s inequality [Top08], one has

diam(!) ( C(n)

(

!!|H|n&1 ,

so that (2.4) and (2.6) imply

diam(!) ( C(n)L . (2.7)

Alternatively, by the monotonicity identity (see [DL08, Theorem 2.1]) and by H ( 2n on !!,one has that

s # (0,') .& e2nsHn(!! ,Bx,s)

snis monotone increasing for every x # Rn+1 . (2.8)

If x # !!, then this function converges to Hn({z # Rn : |z| < 1}) as s & 0+, and thus oneobtains the uniform lower perimeter estimate

Hn(!! ,Bx,s) ! c(n) sn , )x # !! , s # (0, 1) . (2.9)

We notice that this last fact can be used jointly with P (!) ( C(n)L to infer (2.7). Finally,whenever ! satisfies (2.1) we define the Heintze-Karcher deficit of ! as

,(!) =

-!!

nH % (n+ 1)|!|-

!!nH

= 1% (n+ 1)|!|-!!

nH

. (2.10)

Just like #(!), this is a scale invariant quantity such that ,(!) = 0 if and only if ! is a ball.One has

,(!) ( #(!) . (2.11)


Indeed,

,(!) = 1% (n+ 1)|!|-!!

nH

=(n+ 1)|!|-

!!nH 0

% (n+ 1)|!|-!!

nH

=(n+ 1)|!|

n

-!!

1H %

-!!

1H 0-

!!1H 0

-!!

1H

( (n+ 1)|!|n

#(!)-!!

1H-

!!1H 0

-!!

1H

= #(!) .

Torsion potential: We denote by f and u the smooth functions defined on ! by setting)#f = 1 in ! ,

f = 0 on !! ,u = %f . (2.12)

Note that f < 0 on !, with *f = |*f | "! on !!, and *&f = "! · *f = |*f | > 0 on !! byHopf’s lemma. We shall use two integral identities involving f , namely, the Reilly’s identity(see, e.g., [Ros87, Equation (3)])

(

!!H |*f |2 =

(

!(#f)2 % |*2f |2 , (2.13)

and the Pohozaev’s identity, see e.g. [AM07, Theorem 8.30],

(n+ 3)

(

!(%f) =

(

!!(x · "!)|*f |2 . (2.14)

The first one quickly leads to prove Alexandrov’s theorem and the Heintze-Karcher inequality,as shown in [Ros87].

Lemma 2.1. If ! and f are as in (2.1) and (2.12), then

|!|n+ 1

!(

!!

n

H% (n+ 1)|!|

"=

(

!!

1

H

(

!|*2f |2 % (#f)2

n+ 1(2.15)

+

(

!!

1

H

(

!!|*f |2H %

!(

!!|*f |

"2,

In particular, (1.14) holds, and if H is constant on !!, then H = H0 > 0 and ! is a ball.

Proof. By the divergence theorem and by Holder’s inequality,

|!|2 =# (

!!*&f

$2=

!(

!!

/H |*f |/

H

"2(

(

!!

1

H

(

!!|*f |2H .

Thanks to (2.13),(

!!H |*f |2 = n

n+ 1|!|+

(

!

(#f)2

n+ 1% |*2f |2 ( n

n+ 1|!| ,

where we have used the Cauchy-Schwartz inequality

(trM)2 = (M : Id)2 ( |Id|2 |M |2 = (n+ 1) |M |2 , )M # Rn 0 Rn .

(Here and in the following, we denote by : the scalar product on Rn 0 Rn, and by | · | thecorresponding Hilbert norm on Rn 0 Rn.) This proves (2.15). Let us now assume that H isconstant on !!, then by applying the divergence theorem on !! and on !, one finds

(

!!

n

H=

nP (!)

H=

-!! div !!(x) dHn

x

H(2.16)

=

-!!(x · "!)H dHn

x

H=

(

!!x · "! dHn

x =

(

!div (x) dx = (n+ 1)|!| ,


so that H = H0 and equality holds in (1.14). In particular, (2.15) gives that *2f(x) = Id/(n+1)for every x # !, so that, being ! connected,

f(x) = c+|x% x0|2

2(n+ 1),

for some c < 0 and x0 # Rn. Since f = 0 on !!, we find that ! is the ball of center x0 andradius

*%2(n+ 1)c. !

We now exploit [Tal76] and [CGS94] to obtain universal estimates on f .

Lemma 2.2. If ! and f are as in (2.1) and (2.12), then

$f$C0(!) ( 1

2(n+ 1)

! |!||B|

"2/(n+1)( C |!|2/(n+1) , (2.17)

$*f$C0(!) (/2 $f$1/2C0(!) ( C0 |!|1/(n+1) , (2.18)

$*2f$L2(!) ( |!|1/2 . (2.19)

Proof. By a classical result of Talenti [Tal76], the radially symmetric decreasing rearrangement(%f)' of %f satisfies the pointwise estimate

(%f)'(x) ( R2 % |x|2

2(n+ 1), where R =

! |!||B|

"1/(n+1), (2.20)

so that the first inequality in (2.17) follows immediately. The second inequality in (2.17) is thenobtained by recalling that

|{z # Rk : |z| < 1}| = -k/2

&(1 + (k/2)), lim

t#$

&(1 + t)/2-t(t/e)t

= 1 . (2.21)

(Thus (2.17) does not need the assumption that H > 0 on !!). Moreover, we immediatelydeduce (2.19) from #f = 1 and (2.13), so that we are left to prove (2.18). With u = %f we set

p = |*u|2 + 2(u% $u$C0(!)) ,

and aim to prove that p ( 0. The key fact is the observation that

|*u|2#p+ 2*u ·*p ! |*p|2

2, on {|*u| > 0} , (2.22)

see [CGS94, Equation (2.7)]. Given (2.22), we argue by contradiction and assume that themaximum p0 of p in ! is positive. We first claim that p0 is achieved on !!. Indeed, letU = {x # ! : p(x) = p0}, then U is obviously closed. If x # U , then |*u(x)|2 ! p(x) = p0 > 0and so p satisfies

#p+ T ·*p ! 0 , in a neighborhood of x ,

where the vector-field

T = 2*u

|*u|2 (2.23)

is bounded on that same neighborhood. By the strong maximum principle, p must be constantin that neighborhood. This shows that U is open, so that U = ! by connectedness. At the sametime, there exists x% # ! such that *u(x%) = 0, so that p0 = p(x%) = 2

#u(x%) % $u$C0(!)

$( 0

a contradiction. This shows that there exists x0 # !! such that

p(x0) = p0 > p(x) , )x # ! .

By Hopf’s lemma, *&u(x0) < 0, so that

#p+ T ·*p ! 0 , in a neighborhood of x0 in ! ,


where once again the vector-field T (defined as in (2.23) above) is bounded. By Hopf’s lemma,

*&p(x0) > 0 . (2.24)

At the same time one has

*&p(x0) = 2!*u(x0) ·*(*&u)(x0) +*&u(x0)

".

Since u = 0 on !!, we have *u = (*&u) " on !!, so that the above identity becomes

*&p(x0) = *&u(x0)#*& &u(x0) + 1

$.

Since *&u(x0) < 0, (2.24) gives us

*& &u(x0) < %1 = #u(x0) . (2.25)

We now obtain a contradiction by showing that, thanks to H > 0, one has

#u(x0) < *& &u(x0) . (2.26)

Indeed, assuming without loss of generality that x0 = 0 and that ! is (locally at 0) the subgraphof a function . on n-variables such that .(0) = 0 and *.(0) = 0 (so that "!(0) = en, and thus%H(0) = #.(0)), by di"erentiating u(z,.(z)) = 0 at z = 0 twice along the direction zi, onegets

0 = *ziziu(z,.(z)) + 2*zi &u(z,.(z))*zi.(z) +*&u(z,.(z))*zi zi.(z) +*&&u(z) (*zi.(z))2

which, evaluated at z = 0, by *.(0) = 0 gives us

0 = *ziziu(0) +*&u(0)*zi zi.(0) .

By adding up over i = 1, ..., n, and by H(0) > 0 and *&u(0) < 0, we conclude that

0 = #u(0)%*& &u(0)%H(0)*&u(0) > #u(0)%*& &u(0) ,

so that (2.26) holds. !Lemma 2.3. If ! and f are as in (2.1) and (2.12), then

C(n) |!|*,(!) !

(

!

+++*2f % Id

n+ 1

+++ . (2.27)

If, in addition, #(!) ( 1/2, then

C(n)! n

H0

"2P (!) #(!) !

(

!!

+++n/H0

n+ 1% |*f |

+++2. (2.28)

Remark 2.4. If we define u = %f on !, u = 0 on Rn+1 \ !, then the distributional gradientDu and the distributional Hessian D2u of u are given by

Du = %*f Ln+1"! ,

D2u = %*2f Ln+1"!+*f 0*f

|*f | Hn"!! ,

where Ln+1 is the Lebesgue measure on Rn+1. Indeed, for every . # C$c (Rn+1) one has

D2u(.) =

(

Rn+1u !ij. = %

(

!f !ij. =

(

!!if !j. =

(

!!("!)j .!if %

(

!.!ijf ,

where "! = *f/|*f | on !!. Hence, under the assumption (2.2), (2.27) and (2.28) are equivalentto

|D2u% µ|(Rn+1) ( C(n)#P (!) #(!) + |!| ,(!)1/2

$( C(n,L) #(!)1/2 , (2.29)

where µ is the Radon measure defined by

µ = % Id

n+ 1Ln+1"!+

"! 0 "!n+ 1

Hn"!! .


This point of view on (2.27)–(2.28) is at the basis of the proof of Theorem 2.5 below.

Proof of Lemma 2.3. We first prove (2.27). If M1,M2 # Rn 0 Rn with M1,M2 += 0, then onehas

|M1| |M2|%M1 : M2 =1

2

+++µM1 %M2

µ

+++2, µ = (|M2|/|M1|)1/2 ,

so that

|M1|2 |M2|2 % (M1 : M2)2 ! (M1 : M2)

+++µM1 %M2

µ

+++2.

By (2.15), setting M1 = *2f (note that *2f += 0 as #f = 1 on !) and M2 = Id, and noticingthat |Id|2 = (n+ 1) and #f = *2f : Id, one finds

n |!| ,(!) !(

!|Id|2 |*2f |2 % (#f)2 !

(

!µ2

+++*2f % Id

µ2

+++2, (2.30)

where we have set µ(x) =#|Id|/|*2f(x)|

$1/2, x # !. By (2.19) and (2.30), we get

!(

!

+++*2f % Id

µ2

+++"2

((

!µ2

+++*2f % Id

µ2

+++2(

!

|*2f ||Id|

( C(n) |!|3/2 ,(!)!(

!|*2f |2

"1/2( C(n) |!|2 ,(!) ,

that is (

!

+++*2f % Id

µ2

+++ ( C(n) |!|*,(!) . (2.31)

In particular, |tr (M1)% tr (M2)| ( |M1 %M2| and #f = 1 give us(

!

+++1%n+ 1

µ2

+++ ( C(n) |!|*,(!) ,

which leads to (

!

+++Id

n+ 1% Id

µ2

+++ =/n+ 1

(

!

+++1

n+ 1% 1

µ2

+++ ( C(n) |!|*,(!) .

We prove (2.27) by combining this last inequality with (2.31). We now prove (2.28). By (2.15)one has

|!|n+ 1

!(

!!

n

H% (n+ 1)|!|

"! 2

(

!!|*f |

!!(

!!

1

H

(

!!|*f |2H

"1/2%

(

!!|*f |

". (2.32)

Since-!! |*f | = |!|, if $ > 0 is such that

$4 =!(

!!

1

H

"&1(

!!|*f |2H ,

then (2.32) gives us

1

2(n+ 1)

!(

!!

n

H% (n+ 1)|!|

"!

!(

!!

1

H

(

!!|*f |2H

"1/2%(

!!|*f |

=

(

!!

$2

2

1

H+

1

2$2|*f |2H % |*f |

=

(

!!

1

2

! $/H

% |*f |/H

$

"2!

(

!!

H

2$2

!$2

H% |*f |

"2.

Again, by (2.6),

H0

$2

(

!!

!$2

H% |*f |

"2( C(n) ,(!)

(

!!

n

H( C(n)P (!)

n

H0,(!) .


Finally, by (2.13) one has-!!H|*f |2 ( |!|, so that

$4 ( C H0|!|

P (!)( C , (2.33)

and thus(

!!

!$2

H% |*f |

"2( C(n)

! n

H0

"2P (!) ,(!) . (2.34)

Now let |*f |!! denote the average of |*f | on !!, so that |*f |!! = |!|/P (!). By (2.33) and(2.34) we thus find

(

!!

!|*f |!! % |*f |

"2(

(

!!

! $2

H0% |*f |

"2( 2

(

!!

! $2

H0% $2

H

"2+ 2

(

!!

!$2

H% |*f |

"2

( C(n)! n

H0

"2P (!) #(!) ,

where in the last inequality we have used (2.11) and #(!) ( 1. We deduce (2.28) by noticingthat |*f |!! = n/(n+ 1)H0. !

We now exploit a compactness argument to show that if the Alexandrov’s deficit of ! issmall enough, then ! can be taken arbitrarily close (in various ways) to a finite family of tangentballs of unit radii.

Theorem 2.5. Given n,L # N, n ! 2, L ! 1, a # (0, 1], and / > 0 there exists c(n,L, a, /) > 0with the following property. If ! satisfies (2.1), (2.2), (2.4), f is defined as in (2.12) (and thenextended to 0 on Rn+1 \ !) and #(!) ( c(n,L, a, /), then there exists a finite family of disjointunit balls {Bzj ,1}j!J with # J ( L such that, setting

G ='

j!JBzj ,1 ,

!G is connected (that is, each sphere !Bzj ,1 intersects tangentially at least another sphere !Bz",1

for some ( += j) and

|!#G|+ hd(!!, !G) + |P (!)% P (G)|+ $f % fG$C0(Rn+1) ( / ,

where

fG(x) = %.

j!Jmax

/1% |x% xj |2

2(n+ 1), 00, x # Rn+1 .

Moreover, there exist $ " !G and 0 # C1,$($) for every ' # (0, 1) such that !G \ $ consistsof at most C(n)L-many spherical caps whose diameters are bounded by / , and such that (Id +0 "G)($) " !! with

$0$C1(") +Hn#!! \ (Id + 0 "G)($)

$( / , $0$C1,!(") ( C(n, ') .

Remark 2.6. Notice that by Theorem 2.5 and since $*f$C0(!) (/2 $f$1/2C0(!) thanks to (2.18),

one can deduce that$f$C1(!) ( C0(n) , (2.35)

whenever #(!) ( c(n,L, a). (Indeed, it is enough to pick / = /(n) and use $f % fG$C0(!) ( / .)As a consequence one can choose, in the proof of Theorem 1.1, if working with (2.17)–(2.18)or with (2.35). In the former case, one obtains larger powers of L but explicitly computableconstants C(n) in the quantitative estimates of Theorem 1.1; in the latter case, we obtain smallerpowers of L but lose the ability of computing the corresponding constants C(n). We shall optfor the second possibility.


Proof of Theorem 2.5. Let us consider a sequence of sets {!h}h!N satisfying (2.1), (2.2) and(2.4) (with the same L and a for every h # N), and correspondingly define fh starting from !h

by (2.12). Assuming that #(!h) & 0, it will su%ce to prove that, up extracting subsequences,

limh#$

|!h#G|+ hd(!!h, !G) + |P (!h)% P (G)|+ $fh % fG$C0(Rn+1) = 0 , (2.36)

where G and fG are associated to a family of balls {Bzj ,1}j!J as in the statement, and thatthere exist $h " !G and 0h # C1,$($h) for every ' # (0, 1) such that !G \ $h consists of atmost C(n)L-many spherical caps with vanishing diameters, and (Id + 0h "G)($h) " !!h with

limh#$

$0h$C1("h) +Hn#!!h \ (Id + 0h "G)($h)

$= 0 , sup

h!N$0h$C1,!("h) ( C(n, ') .

To this end, we first note that, by (2.7), up to translating the sets !h one has

!h " BR , )h # N , (2.37)

where R = R(n,L). By (2.37) and since P (!h) ( C(n,L) thanks to (2.4), the compactnesstheorem for sets of finite perimeter [Mag12, Theorem 12.26] implies that, up to extractingsubsequences,

limh#$

|!h#G| = 0 , (2.38)

where G " BR is a set of finite perimeter in Rn+1. Similarly, if we define uh : Rn+1 & R bysetting uh = %fh on !h, and uh = 0 on Rn+1 \ !h, then by (2.18) and by (2.37) we find that,again up to extracting subsequences,

limh#$

$uh % u$C0(Rn+1) + $uh % u$L1(Rn+1) = 0 , (2.39)

where u : Rn+1 & [0,') is a Lipschitz function on Rn+1. Now, by Remark 2.4,

D2uh = %*2fh Ln+1"!h + |*fh| "!h 0 "!h Hn"!!h . (2.40)

In particular,

|D2uh|(Rn+1) =

(

!h

|*2fh|+(

!!h

|*fh|

( |!h|1/2$*2fh$L2(!h) +!(

!!h

1

H

"1/2!(

!!h

H |*fh|2"1/2

( |!h|+ C(n)P (!h)1/2 |!h|1/2 ( C(n,L) ,

where in the last line we have used, in the order, (2.19), (2.6), (2.13), (2.5) and (2.4); as aconsequence,

Du # BV (Rn+1;Rn+1) , D2uh%1 D2u as Radon measures on Rn+1 . (2.41)

If . # C0c (Rn+1), then by (2.27) and (2.38)

(D2uh"!h)(.) = %(

!h

.*2fh & % Id

n+ 1

(

G. ,

so that

D2uh"!h%1 % Id

n+ 1Ln+1"G as Radon measures in Rn+1 . (2.42)

By (2.41) and (2.42), if µ denotes the weak-* limit of the Radon measures

µh = D2uh"(Rn+1 \ !h) = |*fh| "!h 0 "!h Hn"!!h ,

then we have

D2u = % Id

n+ 1Ln+1"G+ µ . (2.43)


We claim that

|{u > 0} \G| = 0 , sptµ , {u > 0} = 1 . (2.44)

To prove the first part of (2.44), we note that if u(x) > 0, then by uniform convergence uh !u(x)/2 on Bx,sx for every h ! hx and for some sx > 0, so that Bx,sx " !h for every h ! hx. Thisimplies that |Bx,sx \G| = 0 (thus the first part of (2.44)), and also that Bx,sx , sptµh = 1: sincesptµ is contained in the set of the accumulation points of sequences {xh}h!N with xh # !!h, wehave proved (2.44). By combining (2.43) and (2.44) we deduce that

D2u"{u > 0} = % Id

n+ 1Ln+1"{u > 0} . (2.45)

Now let {Aj}j!J denote the connected components of the open set {u > 0}, then by (2.45) wecan find zj # Rn+1 and cj # R such that

u(x) = cj %|x% zj |2

2(n+ 1), )x # Aj ,

and since u ! 0 it must be

cj ! 0 , Aj " Bzj ,sj where sj = (2(n+ 1)cj)1/2 ,

thus cj > 0 because Aj is open. In conclusion,

{u > 0} ='

j!JAj "

'

j!jBzj ,sj " {u > 0} ,

that is, u = %fG,

u(x) =.

j!Jmax

/s2j % |x% zj |2

2(n+ 1), 00, Aj = Bzj ,sj . (2.46)

We now want to prove that |G#{u > 0}| = 0 and that J is finite with sj = 1 for every j # J .To this end we first notice that sj ( 1 for every j # J . Indeed, by (2.46) we have that

{u > 2} ='

j!JB

zj ,!

(s2j&2(n+1)()+, )2 > 0 ,

so that, by uniform convergence,'

j!JB

zj ,!

(s2j&2(n+1)()+"

%uh >

2

2

&" !h , )h ! h( .

In particular, if we fix j # J , pick 2 < s2j/2(n + 1), and let h ! h(,j , then by the previousinclusion there exists y # !!h such that

n1s2j % 2(n+ 1)2

! H!!h(y) ! n(1% #(!h)) ,

that is, letting h & ', s2j % 2(n+ 1)2 ( 1. By the arbitrariness of 2, we conclude that sj ( 1.We now apply Pohozaev’s identity (2.14) to fh to find

(n+ 3)

(

Rn+1uh = (n+ 3)

(

!h

(%fh) =

(

!!h

(x · "!h)|*fh|2 ,

so that by (2.39), (2.28), and the divergence theorem we find

(n+ 3)

(

Rn+1u = lim

h#$

(

!!h

(x · "!h)

(n+ 1)2=

|G|n+ 1

! |B|n+ 1

.

j!Jsn+1j .


At the same time, by (2.46) and a simple computation we find

(n+ 1) (n+ 3)

(

Rn+1u = |B|

.

j!Jsn+3j ,

so that .

j!Jsn+1j (1% s2j ) ( 0 .

Since sj # (0, 1] for every j # J , we conclude that sj = 1 for every j # J . As a consequence,# J ( L, because of

(L+ 1% a)|B| ! limh#$

|!h| = |G| ! |{u > 0}| = #J |B| .

Since J is finite we deduce from (2.46) that

D2u = % Id

n+ 1Ln+1"

'

j!JBzj ,1 +

.

j!J

"Bzj ,10 "Bzj ,1

n+ 1Hn"!Bzj ,1 .

By comparing this formula with (2.43) we conclude that |G#{u > 0}| = 0, provided we canshow that the measure µ appearing in (2.43) is singular with respect to Ln+1, of course. Tothis end, it su%ces to consider the multiplicity one varifolds Vh associated to !!h. Since (in thenotation and terminology of [Sim83, Chapter 8]) the varifolds {Vh}h!N have uniformly boundedmasses (as M(Vh) = Hn(!!h)) and uniformly bounded generalized mean curvatures (thanksto (2.6)), by [Sim83, Theorem 42.7, Remark 42.8] there exists an integer multiplicity rectifiable

n-varifold V such that Vh%1 V as varifolds. In particular, if V is supported on the n-rectifiable

set M , and if 3 denotes the integer multiplicity of V , then, denoting by "M a Borel vector-fieldsuch that "M (x)' = TxM for Hn-a.e. x # M , we get

(

M.3 "M 0 "M dHn = lim

h#$

(

!!h

."!h 0 "!h dHn , ). # C0

c (Rn+1) .

Hence, by (2.28) and by definition of µh and µ we conclude that

µ =3

n+ 1"M 0 "M Hn"M .

As explained this shows that |G#{u > 0}| = 0, and thus, from now one we directly set

G ='

j!JBzj ,1 .

Let us prove that P (!h) & P (G). By the divergence theorem,

++(n+ 1)|!h|% P (!h)++ =

+++(

!!h

!1% H!!h

n

"(x · "!h)

+++ ( diam(!h) #(!h) ,

while at the same time (n+ 1)|G| = P (G), so that

|P (!h)%P (G)| ( (n+1)||!h|%|G||+diam(!h) #(!h) ( (n+1)|!h#G|+diam(!h) #(!h) , (2.47)

and P (!h) & P (G), as claimed. This last fact implies in particular that

Hn"!!h%1 Hn"!G as Radon measures in Rn+1 . (2.48)

By (2.48), (2.9) and a classical argument we immediately prove that hd(!!h, !G) & 0. Since!!h is connected for every h, hd(!!h, !G) & 0 implies that !G is connected. We are thus leftto prove the existence of sets $h and maps 0h with the claimed properties. To this end we putthe proof of the theorem on hold, and recall some basic useful facts from the regularity theoryfor integer rectifiable varifolds. !


Given x # Rn+1, " # Sn and r > 0 we set

C&x,r =

%y # Rn+1 : |p&(y % x)| < r , |(y % x) · "| < r

&, Cr = Cen

0,r , C = C1 ,

D&x,r =

%y # Rn+1 : |p&(y % x)| < r , (y % x) · " = 0

&, Dr = Den

0,r , D = D1 ,

where p&(v) = v% (v · ")" for every v # Rn+1. Given u # Ck,$(Dr), it will be useful to consider,along with the standard Ck,$-norms on Dr, the scaled norms

$u$%Ck,!(Dr)=

k.

j=0

rj&1 $Dju$C0(Dr) + rk&1+$ [Dku]C0,!(Dr) ,

which are invariant by scaling in the sense that, if we set $r(u)(x) = r&1 u(r x) for x # D, then

$$r(u)$Ck,!(D) = $$r(u)$%Ck,!(D) = $u$%Ck,!(Dr), )r > 0 .

We shall need the following technical lemma, which just amounts to a simple application ofthe implicit function theorem, and whose proof can be found in [CLM14, Lemma 4.3]. In thestatement, given u : D4r & R with |u| < 4r on D4r, we set

&r(u) = (Id + u en)(D4r) " C4r .

Lemma 2.7. Given n ! 1, M > 0 and ' # [0, 1] there exist positive constants )0 = )0(n,M, ') <1 and )1 = )1(n,M, ') with the following property. If u1 # C2,1(D4r), u2 # C1,$(D4r), and

maxi=1,2

$ui$%C1(D4r)( )0 , max

%$u1$%C2,1(D4r)

, $u2$%C1,!(D4r)

&( M ,

then there exists & # C1,$(C2r , &r(u1)) such that

Cr , &r(u2) " (Id + &")(C2r , &r(u1)) " &r(u2) ,

$&$C0(C2r(#r(u1))

r+ $*&$C0(C2r(#r(u1)) + r$ [*&]C0,!(C2r(#r(u1)) ( )1 ,

$&$C0(C2r(#r(u1))

r+ $*&$C0(C2r(#r(u1)) ( )1 $u1 % u2$C1(D4r) .

Here, " # C1,1(&r(u1);Sn) is the normal unit vector field to &r(u1) defined by

"(z, u1(z)) =(%*u1(z), 1)*1 + |*u1(z)|2

, )z # D4r .

Next, let S be a Hn-rectifiable set in Rn+1 with bounded generalized mean curvature insome open set V , that is, there exists H # L$(V ;Hn"S) such that

(

Sdiv S X dHn =

(

SX ·H dHn , )X # C1

c (V ;Rn+1) ,

and assume that S = spt(Hn"S), i.e., Hn(S ,Bx,r) > 0 for every x # S, r > 0. Set

+(S, x, r) = r $H$L!(Bx,r;Hn"S) +max/Hn(S ,Bx,r)

4n rn% 1, 0

0, x # S , r > 0 , (2.49)

where 4n = Hn(B , {x1 = 0}). Then for every ' # (0, 1), Allard’s regularity theorem [All72](as stated in [Sim83, Theorem 24.2] – see also [DL12, Theorem 3.2]) gives us positive constants+0(n, ') < 1 and C(n, ') with the following property:

Allard’s theorem: With S as above, if x # S and r > 0 are such that Bx,r "" V and

+(S, x, r) ( +0(n, ') , (2.50)

then there exist " # Sn and a Lipschitz map u : (x+ "') & R with u(x) = 0 such that

S ,C&x,%0 r =

%z + u(z)" : z # D&

x,%0 r

&, $u$%C1,!(D#

x,$0 r)( C(n, ')+(S, x, r)1/4n .


x

$

Bz2,1

Cµx

x,c0"2

Bz1,1

z2z1

Figure 2. If x # $" , !Bzj ,1, then !G , Cµx

x,c0(n)"2 = !Bzj ,1 , Cµx

x,c0(n)"2 , see also

(2.62). Here µx = "G(x) = "Bz1,1(x).

(Note that this statement is a particular case of Allard’s theorem in the sense that we consideronly density one varifolds and we restrict to the codimension one case.) In the following we shallapply this theorem with ' = 1/4n. Correspondingly, we simply set

+0(n) = +0!n,

1

4n

"< 1 .

We now prove a technical lemma which will be useful in the proof of Theorem 1.1 too.

Lemma 2.8. There exist positive constants $(n) < 1 and c0(n) with the following property. Let! satisfy (2.1), (2.2), and (2.4), let {Bzj ,1}j!J be a disjoint family of unit balls, and set

G ='

j!JBzj ,1 , $) = !G \

'

j,*!J,j )=*

B(zj+z")/2,) $ > 0 . (2.51)

Assume that to each $ ( $(n) and x # $) one can associate 5x # (0, 1) and y # !! in such away that

c0(n)$2

2( 5x ( c0(n)$

2 , (2.52)

|x% y| = dist(x, !!) ( +0(n)52x2

, (2.53)

+(!!, y, 5x) ( +0(n)$(n) , (2.54)

|!#G| ( C(n) 5n+1x +(!!, y, 5x)

1/4n . (2.55)

Then for every $ ( $(n) there exists &) : $) & R such that

$&)$C1,!("%) ( C(n, ') , )' # (0, 1) , (2.56)

$&2 $&)$C0("%) + $*&)$C1("%) ( C(n) maxx!"%

+(!!, y, 5x)1/4n , (2.57)

(Id + &)"G)($)) " !! . (2.58)

Remark 2.9. Note that we do not assume !G to be connected. In other words, the balls Bzj ,1

need not to be tangent, although the may be arbitrarily close or even mutually tangent, andactually this last case will somehow be the “worst” case to keep in mind. We also note thatfor $(n) small enough, if $ ( $(n), then !G \ $) consists of finitely many (precisely, at mostC(n)# J-many) spherical caps of diameters bounded by C $.


Proof of Lemma 2.8. We claim that for every $ ( $(n) and x # $) there exists

&x # C1,1/4n(Cµx

x,%0 +x/4, !G)

such that

Cµx

x,%0 +x/8, !! " (Id + &x"G)(C

µx

x,%0 +x/4, !G) " Cµx

x,%0 +x/2, !! ,

$&x$C1,1/4n(Cµxx,$0 &x/4(!G) ( C(n) , (2.59)

5&1x $&x$C0(Cµx

x,$0 &x/4(!G) + $*&x$C0(Cµxx,$0 &x/4(!G) ( C(n)+(!!, y, 5x)

1/4n .

Postponing for the moment the proof of the claim, let us show how it allows one to completethe proof of the lemma. Indeed, (2.59) implies that, for every x1, x2 # $), &x1 = &x2 on theintersection of their respective domains of definition. Then, by setting

&)(z) = &x(z) , )z # Cµx

x,%0 +x/4, !G , x # $) ,

one defines a function &) # C1,1/4n($)) such that (2.56), (2.57) and (2.58) hold. Moreover,(2.56) follows from elliptic regularity, as $&)$C1,1/4n("%)

( C(n) and the mean curvature of the

graph of &) over $) is the mean curvature of !, and thus it is bounded and continuous.We are thus left to prove our claim. To this end we consider r(n) > 0 such that

Hn(Bz,s , !B) ( (1 + C(n) s2)4n sn , )z # !B , )s < r(n) , (2.60)

sup%|(p% z) · "B(z)| : p # Bz,s , !B

&( C(n) s2 , )z # !B , )s < r(n) , (2.61)

we fix x # $), $ ( $(n), let y and 5x be as in the statement, and set

µx = "G(x) = "Bzj ,1(x) for the unique j # J such that x # !Bzj ,1 .

If c0(n)$(n)2 ( r(n), then by definition of µx there exist C(n) and a Lipschitz map wx :(x+ µ'

x ) & R such that

!G ,Cµx

x,c0(n))2 = !Bzj ,1 ,Cµx

x,c0(n))2 =%z + wx(z)µx : z # Dµx

x,c0(n))2

&,

$wx$C2,1(Dµxx,c0(n)%2

) ( C(n) , $wx$%C1(Dµxx,r)

( C(n) r , )r ( c0(n)$2 .

(2.62)

By (2.54) and by Allard’s theorem, there exist "x # Sn and a Lipschitz map ux : (y + "'x ) & Rsuch that ux(y) = 0,

!! ,C&xy,%0 +x =

%z + ux(z)"x : z # D&x

y,%0 +x

&,

$ux$%C1,1/4n(D#xy,$0 &x )

( C(n)+(!!, y, 5x)1/4n .

(2.63)

Now we let

Ky =%z # C&x

y,%0 +x : (z % y) · "x ( 0&, Kx =

%z # Cµx

x,%0 +x : (z % x) · µx ( 0&.

Up to switching "x with %"x, and since |ux| ( C(n) 5x +(!!, y, 5x)1/4n on D&xy,%0 +x by (2.63) we

can assume that ++(Ky#!) ,C&xy,%0 +x

++ ( C(n) 5n+1x +(!!, y, 5x)

1/4n . (2.64)

Similarly, by (2.62) we have |wx| ( C(n) 52x on Dµxx,%0 +x , and thus

++(Kx#G) ,Cµxx,%0 +x

++ ( C(n) 5n+2x ( C(n) 5n+1

x +(!!, y, 5x)1/4n , (2.65)

as (2.49) and (2.6) imply 5x ( C(n)+(!!, y, 5x). Since |y % x| ( +0 5x/2 by (2.53) and 5x < 1,we find

Bx,%0 +x/2 " By,%0 +x " C&xy,%0 +x , as well as Bx,%0 +x/2 " Cµx

x,%0 +x of course ,


and thus, by (2.64) and (2.65),

|!#G| ! |(!#G) ,Bx,%0 +x/2| ! |(Kx#Ky) ,Bx,%0 +x/2|% C(n) 5n+1x +(!!, y, 5x)

1/4n

!++#(Ky + x% y)#Kx

$,Bx,%0 +x/2

++ (2.66)

%C(n) 5n+1x +(!!, y, 5x)

1/4n %++#Ky + x% y

$#Ky

++ .

On the one hand, for every z # Rn+1, r > 0 and ", " * # Sn one has+++!%

p # C&z,r : (p%z) ·" ( 0

&#%p # C&"

z,r : (p%z) ·" * ( 0&"

,Bz,r/2

+++ ! c(n) |"%" *| rn+1 ; (2.67)

on the other hand, again by |y % x| ( +0 52x/2,

|Ky#(x% y +Ky)| ( C(n)P (Ky) |y % x| ( C(n) 5nx |y % x| ( C(n) 5n+2x . (2.68)

By (2.66), (2.67), and (2.68) we conclude that

c(n) |"x % µx| 5n+1x ( C(n) 5n+1

x +(!!, y, 5x)1/4n + |!#G| ,

so that (2.55) and (2.54) give us

|"x % µx| ( C(n)+(!!, y, 5x)1/4n . (2.69)

By (2.53), (2.63), and (2.69), provided $(n) is small enough, there exist a constant C%(n) and aLipschitz map vx : (x+ µ'

x ) & R such that

!! ,Cµx

x,%0 +x/2=

%z + vx(z)µx : z # Dµx

x,%0 +x/2

&,

$vx$C1,1/4n(Dµxx,$0 &x/2)

( C%(n) , $vx$%C1(Dµxx,$0 &x/2

) ( C%(n)+(!!, y, 5x)1/4n .

By this last property, by (2.52), and by (2.62) we can apply Lemma 2.7 into the cylinderCµx

x,%0 +x/2: indeed, setting by a rigid motion x = 0 and µx = 0, and choosing

4 r =+05x2

, u1 = wx , u2 = vx , ' =1

4n, M = max{C(n), C%(n)} ,

we find that

maxi=1,2

$ui$%C1(D4r)= max

%$wx$%C1(Dµx

x,$0 &x/2), $vx$%C1(Dµx

x,$0 &x/2)

&

( max%C(n)

+0 5x2

, C%(n)+(!!, y, 5x)1/4n

&( C(n)$(n)1/4n ( )0

#n, ',M

$,

provided $(n) is small enough. By Lemma 2.7, there exists &x # C1,1/4n(Cµx

x,%0 +x/4, !G)

satisfying (2.59). !Proof of Theorem 2.5, conclusion. We now conclude the proof of Theorem 2.5. Let us recall thesituation we left: we have {!h}h!N satisfying (2.1), (2.2) and (2.4) (with the same L and a) and

limh#$

|!h#G|+ hd(!!h, !G) + |P (!h)% P (G)| = 0 , (2.70)

where G is the union over a finite family of disjoint unit balls {Bzj ,1}j!J . To complete the proofof the theorem, we need to prove the existence of $h " !G and of 0h : $h & R such that !G\$h

consists of at most C(n)L-many spherical caps with vanishing diameters, (Id + 0h "G)($h) "!!h, and

limh#$

$0h$C1("h) +Hn#!!h \ (Id + 0h "G)($h)

$= 0 , sup

h!N$0h$C1,!("h) ( C(n, ') , (2.71)

for every ' # (0, 1). To this end, we want to apply Lemma 2.8 to ! = !h. Let us fix $ ( $(n),define $) as in (2.51), and for every x # $) let us set

5x = c0(n)$2 ,


so that (2.52) holds trivially. Let us now fix x # $), and consider yh # !!h such that |x% yh| =dist(x, !!h). By hd(!!h, !G) & 0 we have

|x% yh| ( hd(!!h, !G) ( +0(n)c0(n)2 $4

2, )h ! h) ,

provided h) # N is large enough; in particular, (2.53) holds for h large enough. Next, we noticethat by (2.48) and |yh % x| & 0 we have

lim suph#$

P (!h;Byh,+x) ( P (G;Bx,+x) ,

so that (2.6), the definition of 5x, and (2.60) (applied with s = c0(n)$2 ( r(n)) give us

lim suph#$

+(!!h, yh, 5x) ( 2n5x +P (G;Bx,+x)

4n 5nx% 1 ( C(n)

!$2 +

P (G;Bx,c0(n))2)

4n (c0(n)$2)n% 1

"

( C(n)$2 ( +0(n)$

2( +0(n)$(n)

2; (2.72)

in particular, (2.54) holds for h large enough. Finally, (2.6) and (2.49) imply +(!!h, yh, 5x) !(n/2)5x ! c(n)$2, thus up to take h) large enough to entail |!h#G| ( C(n)$n+3 we find that(2.55) holds. By Lemma 2.8 we conclude that for every $ ( $(n) there exists {&)

h}h+h%"

C1,$($)) for every ' # (0, 1) such that

(Id + &)h"G)($)) " !!h , $&)

h$C1,!("%) ( C(n, ') , $&)h$C1("%) ( C(n)$1/2n , (2.73)

where in proving the last bound we have also taken into account the first inequality in (2.72).Since

Hn#!!h \ (Id + &)

h "G)($))$

( P (!h)% P (G) +Hn(!G \ $))

+++Hn($))%Hn

#(Id + &)

h "G)($))$++ ,

by P (!h) & P (G) and by $&)h$C1("%) ( C(n)$1/2n we find that

lim suph#$

Hn#!!h \ (Id + &)

h "G)($))$( Hn(!G \ $)) + C(n)Hn($))$

1/2n .

We complete the proof of the theorem by first considering any $h & 0, and then by setting0h = &)h

k(h) for a properly chosen k(h) & '. !

We now begin the proof of Theorem 1.1, that is, we consider the problem of turning thequalitative information provided in Theorem 2.5 into quantitative estimates in terms of #(!).Recall that, as in the introduction, we set

% =1

2(n+ 2).

Proof of Theorem 1.1. Step one: With f as in (2.12), let us set

2 = |!|1/(n+1) ,(!)" , !( = {x # ! : dist(x, !!) > 2} , f( = f 6 w( , (2.74)

where w((x) = 2&(n+1)w(x/2) for w # C$c (B) with w ! 0, w(x) = w(%x) for every x # Rn+1,

and-Rn+1 w = 1. We claim that, if C0(n) is the constant appearing in (2.35), then

$*f($C0(!') ( C0(n) , (2.75)

$f( % f$C0(!') ( C0(n) 2 , (2.76)22*2f( %

Id

n+ 1

22C0(!')

( C(n) ,(!)" , (2.77)

$*2f($C0(!') ( C(n) , *2f((x) !Id

2(n+ 1), )x # !( . (2.78)


Indeed, (2.75) and (2.76) are obvious. If x # !(, then (2.77) follows by (2.27) and (1.4),

++*2f((x)%Id

n+ 1

++ ($w$C0(Rn+1)

2n+1

(

!

+++*2f % Id

n+ 1

+++ ( C(n) ,(!)(1/2)&"(n+1) .

Finally, (2.78) follows from (2.77) and (2.11) provided #(!) ( c(n) for c(n) small enough.

Step two: Next we define

5 = C0(n) 2 = C0(n) |!|1/(n+1) ,(!)" , A = {f( < %35} . (2.79)

We claim that

{f < %45} " A " {f < %25} , {f < %5} " !( , (2.80)

and that if {Ai}i!I are the connected components of A, then each Ai is convex and there existxi # Ai and 0 < ri1 ( ri2 < ' such that

Bxi,ri1" Ai " Bxi,ri2

, 1 ! ri1ri2

! 1% C(n) ,(!)" ! 1

2, (2.81)

ri1 ( 1 + C1 #(!) , ri2 ( 1 + C(n) #(!)" , (2.82)(

!\"

i#I Bxi,ri1

(%f) ( C(n) |!|(n+2)/(n+1) ,(!)" . (2.83)

Let us first prove that {f < %5} " !(: indeed, if f(x) < %5 but there exists y # !! with|y % x| = dist(x, !!) ( 2, then the segment joining x to y is contained in !, and thus by (2.35)

%5 > f(x) ! f(y)% C0(n) |x% y| ! f(y)% 5 = %5 ,

a contradiction. Similarly, our choice of 5 and (2.76) imply the other inclusions in (2.80). By(2.78), A = {f( < %35} is an open set with convex connected components {Ai}i!I . Let xi # Ai

be such that f((xi) ( f((x) for every x # Ai, and define

gi(x) =|x% xi|2

2(n+ 1)+ f((xi) , x # Rn+1 .

By (2.77), ++*2(f( % gi)(x)++ ( C(n) ,(!)" , )x # !( , (2.84)

so that, by the convexity of Ai, gi(xi) = f((xi), and *gi(xi) = *f((xi) = 0,

|*f((x)%*gi(x)| ( C(n) ,(!)" |x% xi| , )x # Ai , (2.85)

|f((x)% gi(x)| ( C(n) ,(!)" |x% xi|2 , )x # Ai . (2.86)

Let now ri1 ( ri2 be such that

ri1 = sup%r > 0 : Bxi,r " Ai

&, ri2 = inf

%r > 0 : Ai " Bxi,r

&.

By definition there are "1, "2 # Sn such that xi + ri1"1 , xi + ri2"2 # !Ai " {f( = %35}: hence,

0 = f((xi + ri2"2)% f((xi + ri1"1) (2.87)

! gi(xi + ri2"2)% gi(xi + ri1"1)% C(n) ,(!)" ((ri1)2 + (ri2)

2)

=(ri2)

2 % (ri1)2

2(n+ 1)% C(n) ,(!)" ((ri1)

2 + (ri2)2) ,

that is, setting t = ri2/ri1 ! 1,

C(n) ,(!)" ! (ri2)2 % (ri1)

2

(ri1)2 + (ri2)

2=

t2 % 1

t2 + 1! t% 1

t= 1% ri1

ri2,


thus proving (2.81). The second inequality in (2.82) follows from (2.81) and (2.11), while thefirst one is proved by noticing that Bxi,ri1

" !, and thus for some x0 # !! one has n/ri1 !H(x0) ! H0(1% #(!)) = n(1% #(!)). Finally, thanks to (2.80) and (2.5),

(

!\A(%f) (

(

{&f"4+}(%f) ( 45|!| ( C(n) |!|(n+2)/(n+1) ,(!)" , (2.88)

while (2.81), tn+1 % 1 ( 2n+1 (t% 1) for t = ri2/ri1 # [1, 2], and (2.35) give us

(

A\"

i#I Bxi,ri1

(%f) ( $f$C0(!) |B|.

i!I(ri2)

n+1 % (ri1)n+1

( C(n) ,(!)" |B|.

i!I(ri1)

n+1

( C(n) ,(!)" |A| ( C(n) |!| ,(!)" ,

where in the last inequality we have used |A| ( |!|. This proves (2.83).

Step three: We show the existence of {Bxj ,sj}j!J " {Bxi,ri1}i!I such that {sj}j!J satisfies

maxj!J |sj % 1|diam(!)

( C(n) |!| #(!)" , (2.89)

and, if G% =,

j!J Bxj ,sj (so that G% " ! by construction), then

|! \G%||!| ( C1(n) diam(!) |!| #(!)" , (2.90)

|P (!)%# J P (B)|P (!)

( C(n) diam(!) |!| #(!)" , (2.91)

# J ( L , # J ( C(n) |!| . (2.92)

(Note that (2.91) implies (1.6) by (2.5) and (2.7).) Having in mind to exploit the Pohozaev’sidentity (2.14) (recall the proof of Theorem 2.5), we first notice that, by the divergence theoremand by (2.35),

+++|!|

n+ 1%

(

!!(x · "!)|*f |2

+++ =+++(

!!(x · "!)

! 1

(n+ 1)2% |*f |2

"+++

( diam(!)! 1

n+ 1+ $*f$C0(!)

" (

!!

+++1

n+ 1% |*f |

+++

( C(n) diam(!)

(

!!

+++1

n+ 1% |*f |

+++ .

By (2.28) and P (!) ( C(n) |!|, we thus find+++|!|

n+ 1%

(

!!(x · "!)|*f |2

+++ ( C(n) diam(!)!P (!)

(

!!

+++1

n+ 1% |*f |

+++2"1/2

( C(n) diam(!)P (!) #(!)1/2 . (2.93)

By (2.76), (2.86), and diam(Ai) ( 2, one has

$f % gi$C0(Ai) ( C(n) |!|1/(n+1) #(!)" ,

thus, by (2.83)+++(

!f %

.

i!I

(

Bxi,r

i1

gi+++ ( C(n) |!|(n+2)/(n+1) #(!)" . (2.94)


With xi + ri1"1 as in (2.87), by definition of 5 and by (2.86) we have

C |!|1/(n+1) #(!)" ! 35 = %f((xi + ri1"1)

! %gi(xi + ri1"1)% C(n) (ri1)2 ,(!)"

= % (ri1)2

2(n+ 1)% f((xi)% C(n) (ri1)

2 ,(!)" ,

that is, since we definitely have ri1 ( 2,

% (ri1)2

2(n+ 1)% f((xi) ( C(n) |!|1/(n+1) #(!)" .

By this last estimate and the definition of gi,(

Bxi,r

i1

(%gi) =

(

Bxi,r

i1

%f((xi)%|x% xi|2

2(n+ 1)

( C(n) |Bxi,ri1| |!|1/(n+1) #(!)" +

(

Bxi,r

i1

(ri1)2 % |x% xi|2

2(n+ 1)

= C(n) |Bxi,ri1| |!|1/(n+1) #(!)" +

|Bxi,ri1|(ri1)2

(n+ 3)(n+ 1).

By combining (2.94) with this last inequality we find(

!(%f) (

.

i!I

|Bxi,ri1|(ri1)2

(n+ 3)(n+ 1)+ C(n) |!|(n+2)/(n+1) #(!)" . (2.95)

By Pohozaev’s identity (2.14), (2.93) and (2.95), and by taking into account that |!|1/(n+1) (C(n) diam(!) and (2.2), we find

diam(!)P (!) #(!)1/2 + |!|(n+2)/(n+1) #(!)" ( C(n) diam(!) |!| #(!)" ,

and thus

|!| (.

i!I|Bxi,ri1

|(ri1)2 + C(n) diam(!) |!| #(!)" . (2.96)

By |!| !3

i!I |Bxi,ri1| we finally get

|B|.

i!I(ri1)

n+1!1% (ri1)

2"( C(n) diam(!) |!| #(!)" . (2.97)

Let us set .(r) = rn+1 (1% r2), r ! 0, and note that

.(r) !

456

57

3

4rn+1 , if 0 ( r ( 1

2 ,

1% r

2n+1, if 1

2 ( r ( 1 .(2.98)

With C1 as in (2.82), let us now set

I% =%i # I : 1 ( ri1 ( 1 + C1 #(!)

"&, I%% =

%i # I :

1

2( ri1 ( 1

&.

Since Bxi,ri1" ! for each i # I and by #(!) ( c(n) we find

# I% ( |!||B| , 0 ! .(ri1) ! %C(n) #(!)" , )i # I% ,


so that%|B|

.

i!I$.(ri1) ( C(n) |!| #(!)" . (2.99)

By combining (2.99) with (2.97) one finds

|B|.

i!I\I$.(ri1) ( C(n) diam(!) |!| #(!)" . (2.100)

Since .(ri1) ! 0 for every i # I \ I%, (2.98) implies that

3

4

.

i!I\(I$,I$$)

|Bxi,ri1| ( |B|

.

i!I\(I$,I$$)

.(ri1) ,

and thus, by (2.100),.

i!I\(I$,I$$)

|Bxi,ri1| ( C(n) diam(!) |!| #(!)" . (2.101)

We now prove that ri1 is close to 1 for every i # I%%. Indeed, by exploiting again the fact that.(ri1) ! 0 for every i # I \ I%, together with (2.100) and (2.98), we find that

1

2n+1

.

i!I$$(1% ri1) ( C(n) diam(!) |!| #(!)" ,

which in particular gives

1 ! ri1 ! 1% C(n) diam(!) |!| #(!)" , )i # I%% . (2.102)

Finally, if we set J = I% 2 I%% and sj = rj1 for j # J , then (2.89) follows from (2.102) and thedefinition of I%, while (2.96), (2.101) and (2.89) give us

|!| (.

j!J|Bxj ,sj | s2j + C(n) diam(!) |!| #(!)"

( (1 + C(n) diam(!) |!| #(!)") |G%|+ C(n) diam(!) |!| #(!)" ,i.e.

|! \G%| ( C(n) diam(!) |!| |G%| #(!)" ( C(n) diam(!) |!|2 #(!)" .This proves (2.90). Now by (2.89) and since sj ! 1/2

|P (G%)% (n+ 1)|G%|| = (n+ 1)|B|.

j!Jsnj |sj % 1| ( C(n) max

j!J|sj % 1| |B|

.

j!Jsn+1j

( C(n) diam(!) |!|2 #(!)" ,so that (2.2) gives us

|P (!)% P (G%)| = |(n+ 1)|!|% P (G%)| ( C(n)++|!|% |G%|

+++ C(n) diam(!) |!|2 #(!)"

( C(n) diam(!) |!|2 #(!)" ,which proves (2.91) as |!| ( C(n)P (!), and since, by an entirely similar argument,

|P (G%)%#J P (B)| ( C(n)P (!) diam(!) |!| #(!)" .We conclude this step by proving (2.92): indeed, by (2.5), (2.7) and (2.89)

(L+ 1% a)|B| ! |!| ! |G%| !#1% C(n,L) #(!)"

$|B|#J ,

and thus we conclude by #(!) ( c(n,L, a).

Step four: We prove that

maxx!!G$ dist(x, !!)

diam(!)( C(n) #(!)" . (2.103)


We first notice that if x0 # !Ai, then by (2.85), Ai " Bxi,ri2and ri2 ( 2, one has

+++*f((x0)%(x0 % xi)

n+ 1

+++ ( C #(!)" ,

so that, by |x0 % xi| ! ri1 ! 1/2, one finds

|*f((x0)| ! c1(n) , )x0 # !Ai .

Let A%i be the set of points x # !( \ Ai such that if x0 # !Ai denotes the projection of x onto

the convex set Ai, then the open segment joining x to x0 is entirely contained in !( (and thusin A%

i : in particular, A%i is connected). By (2.78), f((x) ( 0 and !Ai " {f( = %35}, we get

35 ! f((x)% f((x0) = *f((x0) · (x% x0) +1

2

( 1

0*2f((tx+ (1% t)x0)[x% x0, x% x0] dt

! |*f((x0)||x% x0|% C2(n) |x% x0|2 !c1(n)

2|x% x0| ,

where we have used the fact that both *f((x0) and x% x0 are orthogonal to !Ai at x0, and wehave assumed that |x% x0| ( c1(n)/2C2(n). If we denote by

Id(X) =%z # Rn+1 : dist(z,X) < d

&, X " Rn+1 , d > 0 ,

the d-neighborhood of a set X, then, setting k0(n) = c1(n)/2C2(n), we obtain

Ik0(n)(Ai) ,A%i " I6+/c1(n)(Ai) .

By connectedness of A%i and by #(!) ( c(n,L), this proves that

A%i " I6+/c1(n)(Ai) .

Since Ai "" !( (thanks to (2.80)), for every x # !Ai, there exists y # !!( such the opensegment joining x and y is entirely contained in A%

i , and the length of this segment is boundedby 65/c1(n), so that

!Ai " I6+/c1(n)(!!() " I(+(6+/c1(n))(!!) .

Step five: We construct a family of disjoint balls {Bzj ,1}j!J such that if we set

G ='

j!JBzj ,1 ,

then|!#G||!| ( C(n) diam(!) |!| #(!)" , maxx!!G dist(x, !!)

diam(!)( C(n) |!| #(!)" . (2.104)

(Note that (2.104) imply (1.5) and (1.7) thanks to (2.5) and (2.7).) Indeed if we set s*j =min{sj , 1}, then {Bxj ,s"j

}j!J is a family of disjoint balls such that G* =,

j!J Bxj ,s"jsatisfies

G* " ! and|! \G*||!| ( C(n) diam(!) |!| #(!)" , maxx!!G" dist(x, !!)

diam(!)( C(n) |!| #(!)" , (2.105)

thanks to (2.90), (2.103) and (2.89). Next, let us fix j0 # J such that 1 > sj0 . By translatingeach xj with j += j0 into

xj = xj + (1% sj0)xj % xj0|xj % xj0 |

,

and setting sj = sj if j += j0, sj0 = 1, xj0 = xj0 , we find that {Bxj ,sj}j!J is a family of disjoint

balls such that G =,

j!J Bxj ,sj satisfies

|!#G||!| ( C(n) diam(!) |!| #(!)" ,

maxx!!G dist(x, !!)

diam(!)( C(n) |!| #(!)" , (2.106)


thanks to (2.105) and (2.89). By iteratively repeating this procedure on each j # J such thatsj < 1 we finally construct a family G with the required properties.

Step six: In this step we complete the proof of Theorem 1.1 up to statements (i) and (ii). To thisend, we want to apply Lemma 2.8 to ! and G. We first notice that for every x # !G, thanks to(1.7), there exists g(x) # !! such that

|x% g(x)| ( C(n) diam(!) |!| #(!)" . (2.107)

(The point g(x) will play the role of y in Lemma 2.8.) Setting Sj = !Bzj ,1 for j # J , we define{rx}x!!G by the rule

rx = sup/r # (0, 2#(!)#) : Bx,r , (!G \ Sj) = 1

0, if j # J , x # Sj , (2.108)

and then set

$% =%x # !G : rx ! #(!)#

&, (2.109)

for some * = *(n) # (0,%) to be suitable chosen later on, see (2.134). With $) defined as in(2.51), see the statement of Lemma 2.8, it is clear that we can choose c3(n) > 0 in such a waythat

$) " $% , for $ = c3(n) #(!)#/2 . (2.110)

In particular, by Remark 2.9 and by (2.92),

!G \ $ consists of at most C(n)#J-many spherical caps

whose diameters are bounded by C(n)$ ( C(n) #(!)#/2 .(2.111)

We now claim that for every x # $), $ as in (2.110), one can find 5x such that,

c0(n)$2

2( 5x ( c0(n)$

2 , (2.112)

+(!!, g(x), 5x) ( +0(n)$(n) , (2.113)

|x% g(x)| ( +0(n)52x2

, (2.114)

|!#G| ( C(n) 5n+1x +(!!, g(x), 5x)

1/4n , (2.115)

where +0(n) and $(n) are as in Lemma 2.8. In proving the claim, the harder task is accommo-dating (2.113), because it requires to control the perimeter convergence of ! to G localized inballs in terms of the Alexandrov’s deficit.

We now prove the claim. First of all we notice that in order to entail (2.112), and thanksto $) " $%, (2.108) and (2.109), it is enough to pick 5x satisfying

c4(n)

2rx ( 5x ( c4(n) rx , (2.116)

for a suitable constant c4 # (0, 1). Next, we notice that by #(!) ( c(n,L) we can entail

supx!"%

rx ( 2 #(!)# ( r%(n) , (2.117)

for an arbitrarily small constant r%(n). Provided r%(n) is small enough, then (2.117), (2.60) and(2.61) give us

P (G;Bx,r) ( (1 + C(n) r2)4n rn , )r < rx , (2.118)

sup%|(p% x) · "G(x)| : p # Bx,r , !G

&( C(n) r2 , )r < rx . (2.119)


(We are going to use this bounds to quantify the size of P (G;Bg(x),+x), see (2.132) below.) ByChebyshev inequality and by (2.90), we can pick 5x satisfying (2.116) and

Hn!(! \G%) , !Bg(x),+x

"( C(n) |!|2 diam(!) #(!)"&# , (2.120)

Hn#(!! 2 !G% 2 !G) , !Bg(x),+x

$= 0 . (2.121)

(Notice that we are using G% in place of G here, because G% is contained in !, and this willsimplify a key computation based on the divergence theorem.) We include a brief justificationof (2.120) for the sake of clarity: let us set

W =%5 #

#c4 rx2

, c4 rx$: Hn

#(! \G%) , !Bg(x),+

$! K(n) |!|2 diam(!) #(!)"&#

&,

then, with C(n) as in (2.90) and for a suitably large value of K(n), we have H1(W ) ((C(n)/K(n)) #(!)# < c4 #(!)#/2 ( c4 rx/2. Now let us consider the open sets

Uj =/y # Rn+1 : |y % zj | < |y % zj" | )j* += j

0, j # J ,

so that Bzj ,1 " Uj for every j # J , and {Uj}j!J is a partition of Rn+1 modulo a Hn-dimensional

set. The boundary of each Uj is contained into finitely many hyperplanes {Lj,i}mj

i=1, wheremj ( # J ( C(n) |!|. Thus

#{(j, i) : j # J , 1 ( i ( mj} ( C(n) |!|2 . (2.122)

We claim the existence of v # Sn and t% # R such that, setting L%j,i = t% v + Lj,i,

Hn(L%j,i , (! \G%)) ( C(n) |!|6 diam(!) #(!)"/2 , (2.123)

|t%| ( #(!)"/2 , (2.124)

Hn#L%j,i , (!! 2 !G%)

$= 0 . (2.125)

To choose v, we let "j,i be a normal vector to Lj,i, and require v # Sn to be such that

|v · "j,i| !c2(n)

|!|2 , )j # J , 1 ( i ( mj . (2.126)

(The existence of such v is deduced by observing that if 3 > 0, then each spherical stripeY ,j,i = {u # Sn : |u · "j,i| < 3} satisfies Hn(Y ,

j,i) ( C(n)3 so that by (2.122)

Hn!Sn \

'

j!J

mj'

i=1

Y ,j,i

"! Hn(Sn)% C(n) |!|2 3 > 0 ,

provided 3 = c(n)/|!|2 for a suitably small value of c(n).) We now find t%. For a constant M(n)to be properly chosen, let us set

Ij,i =/t # R : |t| < #(!)"/2 ,Hn

#(! \G%) , (t v + Lj,i)

$! M(n) |!|6diam(!) #(!)"/2

0.

If (j,i(y) = y · "j,i then Lj,i = {(j,i = *j,i} for some *j,i, while t v + Lj,i = {(j,i = *j,i + t v · "j,i}.By (2.90) and Fubini’s theorem we find

C1(n) |!|2 diam(!) #(!)" ! |! \G%| =(

RHn((! \G%) , {(j,i = s}) ds

! |v · "j,i|(

RHn((! \G%) , {(j,i = *j,i + t v · "j,i}) dt

! |v · "j,i|H1(Ij,i)M(n) |!|6diam(!) #(!)"/2 ,


that is, by (2.126),

|!|2H1(Ij,i) (C1(n) #(!)"/2

c2(n)M(n), )j # J , 1 ( i ( mj .

By combining this estimate with (2.122), we see that if M(n) is large enough, then

H1!(%#(!)"/2, #(!)"/2) \

'

j!J

mj'

i=1

,Ij,i"> 0 ,

that is, there exists t% such that (2.123), (2.124) and (2.125) hold. If we set U%j = t% v+Uj , then

{U%j }j!J is a partition of Rn+1 modulo a Hn-dimensional set such that !U%

j is contained into

the hyperplanes {L%j,i}

mj

i=1 and such that

Hn#!U%

j , (! \G%)$

( C(n) |!|6 diam(!) #(!)"/2 , (2.127)

Hn#!U%

j , (!! 2 !G%)$

= 0 , (2.128)

Hn#(U%

j , !G%)#Sj$) ( C(n) #(!)"/2 . (2.129)

Here, (2.127) and (2.128) are immediate from (2.123) and (2.125). To prove (2.129), let us recallthat Sj = !Bxj ,sj " Uj , so that by translating the boundary hyperplanes of Uj by t% v with

|t%| ( #(!)"/2 we have possibly cut out from Sj at most mj-many spherical caps of Hn-measurebounded above by

C(n) |t%|n/2 ( C(n) #(!)n"/4 ,

that is, thanks also to mj ( L,

Hn((U%j , Sj)#Sj) ( C(n)L #(!)n"/4 ( C(n) #(!)"/2 ,

thanks to #(!) ( c(n,L). By a similar argument, since Hn(Sj" , Uj) = 0 for j += j*, we havethat Hn(U%

j , Sj") ( C(n) #(!)"/2, and thus (2.129) is proved.We now apply the divergence theorem to the vector field y .& (y% zj)/|y% zj | on the set of

finite perimeter (! \ G%) , (U%j \ Bg(x),+x). Since div ((y % zj)/|y % zj |) = n/|y % zj | for y += zj

and G% " !, by exploiting [Mag12, Theorem 16.3], one finds

0 <

(

(U$j \Bg(x),&x )(!!

y % zj|y % zj |

· "!(y) dHny %

(

(U$j \Bg(x),&x )(!G$

y % zj|y % zj |

· "G$(y) dHny

+

(

!(U$j \Bg(x),&x )((!\G$)

y % zj|y % zj |

· "U$j \Bg(x),&x

(y) dHny

( P (!;U%j \Bg(x),+x) +Hn

!(!U%

j 2 !Bg(x),+x) , (! \G%)"

%P (G%;U%j \Bg(x),+x) + C(n) #(!)"/2

where in the last inequality we have used "G$(y) · [(y % zj)/(y % zj)] = 1 if y # Sj and (2.129).By combining this last inequality with (2.120) and (2.127) we thus find

P (G%;U%j \Bg(x),+x) ( P (!;U%

j \Bg(x),+x)

+C(n) |!|2 diam(!)!#(!)"&# + |!|4 #(!)"/2

".

By adding up over j # J , and since # J ( C(n) |!|, we thus find

P (G%;Rn+1 \Bg(x),+x) ( P (!;Rn+1 \Bg(x),+x)

+C(n) |!|3 diam(!)!#(!)"&# + |!|4 #(!)"/2

",


which gives us, keeping in mind the construction used in step five to define G starting from G%,and also thanks to (2.89) and #(!) ( c(n),

P (G;Rn+1 \Bg(x),+x) ( P (!;Rn+1 \Bg(x),+x) (2.130)

+C(n) |!|3 diam(!)!#(!)"&# + |!|4 #(!)"/2

",

which, combined with (1.6) and (2.121), gives us

P (!;Bg(x),+x) = P (!)% P (!;Rn+1 \Bg(x),+x) (2.131)

( P (G)% P (G;Rn+1 \Bg(x),+x) + C(n) |!|3 diam(!)!#(!)"&# + |!|4 #(!)"/2

"

= P (G;Bg(x),+x) + C(n) |!|3 diam(!)!#(!)"&# + |!|4 #(!)"/2

".

By (2.107), (2.5), (2.7), and thanks to #(!) ( c(n,L), we entail Bg(x),+x " Bx,rx , so that, bydefinition of rx,

P (G;Bg(x),+x) = Hn(Bg(x),+x , !G) = Hn(Bg(x),+x , Sj)

( (1 + C(n) 5x)Hn(Bx,+x , Sj) = (1 + C(n) 5x)P (G;Bx,+x) .

By combining this inequality with (2.131), (2.118), (2.109) and (2.116) we find

P (!;Bg(x),+x)

4n 5nx% 1 ( C(n) 5x +

C(n)

5nx|!|3 diam(!)

!#(!)"&# + |!|4 #(!)"/2

"(2.132)

( C(n)!#(!)# + |!|3diam(!)

##(!)"&(n+1)# + |!|4#(!)("/2)&n#

$".

By combining (2.132) with (2.49) (the definition of +(!!, g(x), 5x)), (2.6) and 5x ( 2#(!)# , wefind

+(!!, g(x), 5x) ( C(n)!#(!)# + |!|3diam(!)

##(!)"&(n+1)# + |!|4#(!)("/2)&n#

$".

For this estimate to be nontrivial we definitely need

% > max{(n+ 1)*, 2n*} = 2n* .

Under this assumption we have %% (n+ 1)* > (%/2)% n*, and thus

+(!!, g(x), 5x) ( C(n)!#(!)# + |!|7 diam(!)#(!)("/2)&n#

"

( C(n) |!|7 diam(!) #(!)"/2(n+1) , (2.133)

where we have set

* =%

2(n+ 1), (2.134)

in order to have * = (%/2)% n*. By #(!) ( c(n,L) and by (2.133), we have thus proved so farthat for every x # $) one can find g(x) # !! and 5x # (0, 1) such that (2.112) and (2.113) hold.

In order to prove our claim, we are left to prove (2.114) and (2.115). By (2.107), (2.5)and (2.7) we have |x% g(x)| ( C(n,L) #(!)" while +0(n)52x/2 ! c(n)#(!)2# by (2.112), so that(2.114) follows by #(!) ( c(n,L) thanks to the fact that % > 2*. Similarly, concerning (2.115)we see by (2.6) that +(!!, g(x), 5x) ! n 5x so that

5n+1x +(!!, g(x), 5x)

1/4n ! c(n) 5n+2x = c(n) #(!)#(n+2) ,

while (1.5) gives us |!#G| ( C(n, L) #(!)", so that #(!) ( c(n,L) implies (2.115) thanks tothe fact that % > *(n+ 2).


We thus proved our claim: for every x # $) there exists g(x) # !! and 5x satisfying(2.112)–(2.115). By (2.110) and #(!) ( c(n,L) we have that $ ( $(n), and thus we can applyLemma 2.8 to find a function & : $) & R such that,

$&$C1,!("%) ( C(n, ') , )' # (0, 1) , (2.135)

$&2$&$C0("%) + $*&$C0("%) ( C(n) maxx!"%

+(!!, g(x), 5x)1/4n , (2.136)

(Id + &"G)($)) " !! . (2.137)

(Notice that (2.135) is (2.56).) Moreover, let us recall from the proof of Lemma 2.8 (see (2.59))that if x # $) , Sj , then the function & is actually defined on Cµx

x,%0 +x/4, !G = Cµx

x,%0 +x/4, Sj ,

withCµx

x,%0 +x/8, !! " (Id + & "G)(C

µx

x,%0 +x/4, !G) " Cµx

x,%0 +x/2, !! ,

$&$C1(Cµxx,$0 &x/4(!G) ( C(n)+(!!, g(x), 5x)

1/4n .(2.138)

Now, by (2.107), #(!) ( c(n,L), and +0 5x/8 ! c(n) #(!)# we find

g(x) # Cµx

x,%0 +x/8, !! ,

and thus, by the first inclusion in (2.138), there exists y # Cµx

x,%0 +x/4, !G such that

g(x) = y + &(y)"G(y) .

By (2.138), (2.133), and #(!) ( c(n,L) we can definitely ensure

$&$C1(Cµxx,$0 &x/4(!G) ( c(n) , (2.139)

so that, taking into account that (2.119) gives |(y % x) · "G(y)| ( C(n) |y % x|2, we find that

|g(x)% x|2 ! |x% y|2 + |&(y)|2 % 2$&$C0(Cµxx,$0 &x/4(!G) |"G(y) · (y % x)| ! |x% y|2

2+ |&(y)|2 .

Again by (2.107) we conclude |x % y| + |&(y)| ( C(n) |!| diam(!) #(!)", and thus, providedc(n) ( 1 in (2.139), |&(x)| ( C(n) |!| diam(!) #(!)". We have thus improved the C0-bound on& in (2.136), by showing that

$&$C0("%) ( C3(n) |!| diam(!) #(!)" . (2.140)

By combining (2.133) with (2.136) and diam(!) ( C(n)P (!) ( C(n)|!|, we obtain

$*&$C0("%) ( C(n) |!|2/n #(!)#/4n . (2.141)

(By (2.140) and (2.141) we deduce (1.10).) Next, by (2.111) and by # J ( C(n) |!| (C(n)P (!), we have

Hn(!G \ $)) ( C(n)P (!) #(!)n#/2 , (2.142)

while by the area formula

Hn#(Id + & "G)($))

$=

(

"%

1(1 + &)2n + (1 + &)2(n&1)|*&|2 ,

so that++Hn($))%Hn

#(Id + & "G)($))

$++ ( P (G)#$&$C0("%) + $*&$2C0("%)

$

( C(n)P (!) |!|4/n #(!)#/2n ,where in the last inequality we have used P (G) ( 2P (!) (which follows by #(!) ( c(n,L) and(2.91)) together with (2.140) and (2.141). By (2.91), (2.142), and

Hn#!! \ (Id + & "G)($))

$

( P (!)% P (G) +Hn(!G \ $)) +++Hn($))%Hn

#(Id + & "G)($))

$++


we thus obtainHn

#!! \ (Id + & "G)($))

$

P (!)( C(n) |!|4/n #(!)#/2n . (2.143)

Now let x0 # !! and / > 0 be such that

/ = maxx!!!

dist(x, !G) = dist(x0, !G) ,

and assume that / ! 2 3 for 3 = C3(n) diam(!) |!| #(!)". By (2.140) and since Bx0,- "Rn+1 \ !G, one has

Bx0,-&, " Rn+1 \ I,(!G) " Rn+1 \ (Id + &"G)($)) ,

so that, thanks to (2.9),

c(n) min{1, / % 3}n ( P (!, Bx0,-&,) ( Hn(!! \ (Id + &"G)($))) ( C(n)P (!) |!|4/n #(!)#/2n .By #(!) ( c(n,L) we thus find min{1, / % 3} = / % 3, and thus

/ ( 3 + C(n)P (!)1/n |!|4/n2#(!)#/2n

2 ( C(n)P (!)1/n |!|4/n2#(!)#/2n

2.

Since P (!) ( C(n)|!| ( C(n) diam(!)n+1 and diam(!) ( C(n)|!|, we find P (!)1/n (C(n) diam(!) |!|1/n and thus conclude that

maxx!!! dist(x, !G)

diam(!)( C(n) |!|3/n #(!)#/2n2

. (2.144)

where have simplified some powers on |!| by noticing that (4/n2)+ (1/n) ( 3/n. By combining(2.104) and (2.144) we obtain (1.8).

Step seven: We complete the proof of the theorem. By contradiction, and by definition of $), if# J ! 2 and dist(Sj , S*) ! 4$ for every ( += j, then Sj " $), and thus &j = (Id+&"G)(Sj) " !!,with &j connected. Since !! is connected, we conclude that !! = &j , thus that # J = 1. Thisproves that if # J ! 2, then for every j # J there is ( += j such that dist(Sj , S*) ( 4$, andthen (i) follows by diam(!) ! c(n) |!|1/(n+1) ! c(n). In order to prove (ii), let us assume that# J ! 2, and let j += ( # J be such that (1.12) holds. Let us set

x =zj + z*

2=

zj + z*2

,

where zj and z* belong to the closed segment joining zj and z* and are such that

!Bzj ,1 , !Bz",1 = {x} .

In this way, for every r # (0,)) and thanks to (1.12) we have

C(n) rn+2 !++Bx,r \

#Bzj ,1 2Bz",1

$++ !++Bx,r \

#Bzj ,1 2Bz",1

$++% C(n) #(!)"/4(n+1) .

By (1.5) we have++Bx,r \

#Bzj ,1 2Bz",1

$++ ! |Bx,r \G| ! |Bx,r \ !|% C(n)L3 #(!)" ,

so that by (1.13) we finally get

) |B| rn+1 ( C(n)#L3#(!)" + #(!)"/4(n+1) + rn+2

$, )r < ) .

We now use this inequality with r = 7), with 7 = 7(n) # (0, 1) to be properly chosen. By#(!) ( c(n,L,)) we can entail,

L3#(!)" + #(!)"/4(n+1) ( 7n+2 )n+2 ,

thus finding )n+2 |B|7n+1 ( C(n)7n+2 )n+2. This is of course a contradiction if we pick 7 smallenough. The proof of Theorem 1.1 is complete. !


3. An application to capillarity-type energies

Proof of Proposition 1.2. It is well-known that (1.19) implies the existence of a constant $ # Rsuch that (

!!div !!X +

(

!g (X · "!) = $

(

!!(X · "!) , )X # C$

c (Rn+1;Rn+1) . (3.1)

Since R0 > 0 is such that ! "" B2R0 , by testing (3.1) with X(x) = .(x)x for some . #C$c (B2R0) with . = 1 on !, one easily obtains that

(n+ 1) |!|$ = nP (!) +

(

!div (x, g(x)) dx . (3.2)

At the same time, since !! is of class C2, (3.1) implies that H + g = $ on !!, which combinedwith (3.2) gives

H(x)%H0

H0=

1

H0

!% g(x) +

-! div (x g(x)) dx

(n+ 1)|!|

"3 #(!) (

C $g$C1(BR0 )

H0.

By the isoperimetric inequality,

H0 =nP (!)

(n+ 1) |!| !n (n+ 1) |B|1/(n+1) |!|n/(n+1)

(n+ 1) |!| = n! |B||!|

"1/(n+1).

so that #(!) ( C(n) $g$C1(BR0 )m1/(n+1), that is (1.20). !

Appendix A. Almost-constant mean curvature implies almost-umbilicality

The purpose of this appendix is to discuss the relation between the Alexandrov’s deficit #(!)and the size of the traceless part A of the second fundamental form A of !!. Having in mindthe quantitative results for almost-umbilical surfaces by De Lellis-Muller [DLM05, DLM06] andPerez [Per11], we seek for a control of A in Lp(!!) for some p ! n. In this direction, we havethe following proposition, where ,(!) denotes the Heintze-Karcher deficit of !, see (2.10).

Proposition A.1. If ! " Rn+1 (n ! 2) is a bounded connected open set with C2-boundary,H > 0 on !!, and #(!) ( 1/2, then

C(n)#P (!) ,(!)

$1/(n+1) $A$Lp( (!!) ! $A$Lp(!!) , )p # [1, n+ 1] , (A.1)

where p' = (n+ 1)p/[(n+ 1)% p] if p < n+ 1, and p' = +' otherwise.

Before proving Proposition A.1, let us discuss (A.1) in connection with the above mentionedresults for almost-umbilical surfaces. Let us consider

3p(!) = inf)!R

$A% $ Id$Lp(!!) , p ! 1 ,

as a measure of the non-umbilicality of !!. In [Per11, Theorem 1.1] it is shown that if ! is abounded connected open set with smooth boundary such that

P (!) = P (B) , $A$Lp(!!) ( K ,

for some p # (n,'), then

inf)!R

$A% $ Id$Lp(!!) ( C(n, p,K) $A$Lp(!!) . (A.2)

Moreover, in [Per11, Corollary 1.2] it is shown that for every 2 > 0 there exists c = c(n, p,K, 2)such that if

$A$Lp(!!) ( c 3 infx!Rn+1

hd(!!, !Bx,1) < 2 . (A.3)


By combining (A.1) and (A.2) with ,(!) ( #(!), we deduce that, if P (!) = P (B), H > 0 and#(!) ( 1/2, then

3p(!) ( C(n, p, $A$Lp( (!!)) #(!)1/(n+1) .

If, in addition, #(!) ( c(n, p, $A$Lp( (!!), 2), then, by (A.3),

infx!Rn+1

hd(!!, !Bx,1) < 2 .

A similar comparison is possible with the results of De Lellis and Muller, which pertain the casen = p = 2. In conclusion, the use of almost-umbilicality in attacking Theorem 1.1 does not seemto provide one with a starting point as e"ective as the one based on the study of the torsionpotential discussed in section 2. We finally prove the above proposition.

Proof of Proposition A.1. This is consequence of the proof of the Heintze-Karcher inequality byMontiel-Ros [MR91], which we now recall for the reader’s convenience. For each x # !! let{)i(x)}ni=1 be the principal curvatures of !! at x, so that, if we set ) = max1"i"n )i, then) ! H/n > 0 on !!. Let us consider the set

& =/(x, t) # !!4 (0,') : t ( 1

)(x)

0,

and the function g : !!4 (0,') & Rn+1

g(x, t) = x% t "!(x) , (x, t) # !!4 (0,') .

We claim that ! " g(&). Indeed, given y # ! let x # !! be such that |x% y| = dist(y, !!). If' is a curve in !! with '(0) = x and '*(0) = / # Sn, then we obtain

(' % y) · '* = 0 , ('(0)% y) · '**(0) + '*(0)2 ! 0 .

In particular, there exists t > 0 such that x = y % t "!(y), and it must be 1 % )(y) t ! 0. Wenow combine ! " g(&) with the area formula, the arithmetic-geometric mean inequality, and) ! H/n, to prove the Heintze-Karcher inequality

|!| ( |g(&)| ((

g(#)H0(g&1(y)) dy =

(

#J# g(x, t) dHn(x) dt

=

(

!!dHn

( 1/.

0

n8

i=1

(1% t)i) dt

((

!!dHn

( 1/.

0

! 1

n

n.

i=1

(1% t)i)"n

dt =

(

!!dHn

( 1/.

0

!1% t

H

n

"ndt

((

!!dHn

( n/H

0

!1% t

H

n

"ndt =

1

n+ 1

(

!!

n

H.

This chain of inequalities implies of course the following identity(

!!

n

HdHn ,(!)

n+ 1=

1

n+ 1

(

!!

n

H

!1% 1

)

H

n

"n+1+

(

#µnA % µn

G (A.4)

+

(

g(#)

#H0(g&1(y))% 1

$dy +

#|g(&)|% |!|

$,

where for every (x, t) # & we have set µi(x, t) = 1% t)i(x) (note that µi ! 0 on &) and

µA =1

n

n.

i=1

µi , µG =n8

i=1

µ1/ni .


The first two terms on the right-hand side of (A.4) provide some control on A. Since |A(x)| (C(n) |)(x)% (H(x)/n)| for every x # !!, by looking at the first term, we get

C(n) ,(!)

(

!!

n

H!

(

!!

n

H

! |A|)

"n+1,

that is, by exploiting #(!) ( 1/2 to infer H0/2 ( H(x) ( 2H0 for every x # !!,

C(n)P (!) ,(!) !(

!!

! |A|)

"n+1,

We thus conclude the proof by Holder inequality. !

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Dipartimento di Matematica e Informatica, Universita di Palermo, Via Archirafi 34, 90123Palermo, Italy

E-mail address: [email protected]

Department of Mathematics, University of Texas at Austin, Austin, TX, USAE-mail address: [email protected]

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Introduction - cvgmt: homecvgmt.sns.it/media/doc/paper/2660/CiraoloMaggi.pdf · Introduction We investigate the geometry of compact boundaries with almost constant mean curvature