Math. Nachr. 279, No. 13–14, 1597 – 1601 (2006) / DOI 10.1002/mana.200310439

B-sub-manifolds and their stability

Ma Li∗1

1 Department of Mathematical Sciences, Tsinghua University, Beijing, P. R. China

Received 28 April 2003, revised 5 May 2006, accepted 17 July 2006Published online 7 September 2006

Key words Minimal sub-manifold, soliton, mean curvature

MSC (2000) 53C42

In this paper, we show that the “grim reaper” in the curve-shortening problem is stable in the sense of “sym-metric stable” defined by K. Smoczyk. We also briefly discuss the graphic B-minimal sub-manifold in R

n+k.

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1 Introduction

The aim of this paper is to answer a question posed by K. Smoczyk [10]. Let (Mm, g) be a Riemannian manifold.Assume that B is a smooth function on Mm. Let N = M × S1 and let ds2 = g + e2Bdt2, where t ∈ S1.

Definition 1 Let Σk be a k-dimensional manifold. Let F : Σ → M be an immersion. We say that ak-dimensional sub-manifold Σk is a B-minimal sub-manifold in M if the immersion

(x, t) ∈ Σ × S1 −→ (F (x), t) ∈ N

is a minimal sub-manifold in

(N, ds2 = g + e2Bdt2

).

Assume that a k-dimensional sub-manifold Σk is a B-minimal sub-manifold in M . Then the volume of

Σ × S1 ⊂ N

is

V (Σ) = 2π

∫Σ

eB(x) dvg

where dvg is the induced volume form in Σk. Hence, a B-minimal sub-manifold is in fact a minimal sub-manifoldin M equipped with the conformal metric e2B/kg.

Definition 2 We say that a B-minimal sub-manifold in M is stable if the second variational derivative of thevolume functional V (Σ) is positive semi-definite.

Remark 1 Let X = gradB be the vector field on M . If X is a Killing or a conformal vector field, then it iseasy to see that the B-minimal submanifolds are the soliton solutions of the mean curvature flow in M as notedearlier by G. Huisken [7], S. Angenent [1], and K. Smoczyk [10]. With this understanding, we can easily get thefirst and second variational formulae for a B-minimal sub-manifold (see [8, 9]).

Our main result is the following

∗ e-mail: lma@math.tsinghua.edu.cn, Phone: +86 10 6278 7513, Fax: +86 10 6278 1785

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1598 Ma Li: Minimal sub-manifolds and their stability

Theorem 1 Assume B(x, y) = y in the xy-plane. Then the grim reaper soliton y = − log cosx in the curve-shortening flow is a B-minimal sub-manifold of dimension one, which is stable in both our sense above and thesymmetric stability defined by K. Smoczyk [10]. In fact, we have the following inequality

∫ π/2

−π/2

3 cos2 x − 14 cosx

u2(x) dx ≤∫ π/2

−π/2

(u′(x))2 cosxdx, (1.1)

where u′ is the derivative of u and u ∈ C∞0

(− π2 , π

2

).

We remark that the inequality above was first posed by K. Smoczyk in [10]. Theorem 1 is proved in thenext section. In the last section, we will discuss the graphic B-minimal sub-manifold in R

n+k and some relatedquestions.

2 Proof of Theorem 1

First of all, let us see why the inequality in Theorem 1 is true from another point of view.Let γ(x) = (x, y(x)) be a smooth curve in the plane R

2. Let B(x, y) = y in R2. Let

w(x) =√

1 + y′2.

We now define a new functional

I(y) =∫ b

a

w(x)ey(x)dx.

It is easy to see that the first variational formula of I(·) is

δI(y)ξ =d

dtI(y + tξ)

∣∣t=0

=∫

[a,b]

{y′ξ′

w(x)+ w(x)ξ(x)

}ey dx.

Here ξ ∈ C10 [a, b]. Hence a critical point y = y(x) of I(·) satisfies

−e−y(x) d

dx

ey(x) y′p

1 + y′2

!+

√1 + y′2 = 0. (2.1)

We remark that a more general equation will be discussed in Section 3.Now we directly compute the second variational formula of I(·) in the following way:

δ2I(y)(ξ, ξ) =d

dtδI(y + tξ)ξ

∣∣t=0

=∫ b

a

ξ′2

w(x)3ey dx ≥ 0.

Hence, any critical point y = y(x) of I(·) is stable in the sense that the second variational derivative at y issemi-positive definite.

Take

y(x) = − log cosx, x ∈ (−π/2, π/2).

The curve (x, y(x)) is called the grim reaper in the curve-shortening problem in the plane. Then one has

−e−y(x) d

dx

ey(x)

y′p1 + y′2

!= − cosx

(cosx tan x × 1

cosx

)′= − 1

cosx= −

√1 + y′2.

Let B(x, y) = y. By definition, this grim reaper is the B-minimal curve in R2. Therefore, it is stable in our sense

above.

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Math. Nachr. 279, No. 13–14 (2006) 1599

Now let us prove Inequality (1.1) directly. Let ε > 0 be a small positive number. We let

p = pε(x) = ε + cosx,

and let J =(− π

2 , π2

). We define a new measure

dµ = p(x) dx

and a new function

f =3p2 − 1

4p.

Then we only need to prove the following inequality:

∫J

fu2 dµ ≤∫

J

(u′)2 dµ. (2.2)

If this is true, we just send ε → 0+ and get Inequality (1.1).We look for a function φ = φ(x) such that

∫J

(u′ − uφ)2 dµ ≤∫

J

((u′)2 − fu2

)dµ, (2.3)

which clearly strengthens the inequality (2.2). Note that this inequality is equivalent to

∫J

(u2

(φ2 + f

) − 2uu′φ)dµ ≤ 0.

By using integration by part, we have

∫J

(2uu′φ) dµ = −∫

u2

(φ′ − φ

sinx

p

)dµ.

Hence Inequality (2.3) can be written as

∫J

u2»φ2 + f + φ′ − φ

sin x

p

–dµ ≤ 0.

We now try to solve the following equation for φ:

φ′ + φ2 − φsinx

p+ f = 0.

Let φ = (log v)′. Then we have the following equivalent equation

v′′ − sin x

pv′ + f(x)v = 0. (2.4)

Imposing the initial conditions

v“−π

2

”= 1 and v′

“−π

2

”= 0,

we can solve Eq. (2.4) (see P. Hartman [6, Chapter IV, Lemma 1.1]). Hence Inequality (2.3) is true, and we haveproved Theorem 1.

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1600 Ma Li: Minimal sub-manifolds and their stability

3 Graphic B-minimal sub-manifolds

In this section, we point out some related equations. Let D ⊂ Rn be a domain of R

n. Write x = (x1, . . . , xn) ∈D. Let y = y(x) ∈ R

k be a vector-valued smooth function. Define the graphic sub-manifold

Σ = {(x, y(x)); x ∈ D} ⊂ Rn+k.

Let F (x) = (x, y(x)) be the graphic mapping in the space Rn × R

k. Let B(y) be a smooth function in Rk.

Then we have

DF (x) = (id, Dxy)and let

w(x) =√

det (δij + 〈Dxiy, Dxjy〉) ,

where 〈·, ·〉 is the inner product in Rk.

Write

gij = δij + 〈Dxiy, Dxjy〉and (

gij)

= (gij)−1.

Define a new functional

I(y) =∫

D

w(x)eB(y(x))dx.

It is easy to see that the first variational formula of I(·) is

δI(y)ξ =d

dtI(y + tξ)

∣∣t=0

=∫

D

{gij 〈Dxiy, Dxjξ〉

w(x)+ w(x)〈DyB, ξ(x)〉

}eB(y) dx.

Here ξ ∈ C10 (D). Hence, Σ is a B-minimal sub-manifold in R

n+k if and only if y = y(x) is a critical point ofI(·). So y = y(x) satisfies

−e−B(y)Dxj

„eB(y) gijDxiy

w(x)

«+ w(x)DyB(y) = 0.

In the special case where k = 1 and B(y) = y, we have

w(x) =√

1 + |� y|2,and the B-minimal equation is

−e−ydiv

0@ey �yq

1 + |� y|2

1A+

√1 + |� y|2 = 0,

which is the equation of the mean curvature flow for translating solitons (see also [1]). A special case of thisequation is Eq. (2.1).

From this generality, one may study the stability, or one may study Bernstein type problem and a priori esti-mates for the graphic B-minimal sub-manifolds. One can also propose a similar concept like weighted harmonicmaps from a Riemannian manifold (M, g) to another Riemannian manifold (N, h) in the following way. Supposethat a smooth function B on N is given. Let u : M → N be a C1 mapping and let

e(u) = |du|2 eB(u)

be its energy density on M . Define the B-energy functional as

E(u) =12

∫M

e(u) dx.

Then we call u a B-harmonic map if it is a critical point of the B-energy functional. One may discuss theLiouville property for B-harmonic maps.

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Math. Nachr. 279, No. 13–14 (2006) 1601

Acknowledgements The author would like to thank Prof. James S. W. Wong for his interest in this work. The author thanksthe referee for helpful suggestions.

References

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