Abstracts
In this note, we consider self-similar immersions of the mean curvature flow and show that a graph solution of the soliton equation, provided it has bounded derivative, converges smoothly to a function which has some special properties (see Theorem 1.1).
soliton; Self-Similar; Mean curvature flow
Nesta nota, consideramos imersões auto-semelhantes do fluxo de curvatura média, e mostramos que uma solução em forma de gráfico da equação de soliton converge diferencialmente, contanto que tenha derivada limitada, para um gráfico cuja função tem propriedades especiais (V. Teorema 1.1).
Soliton; Auto-similar; Fluxo de curvatura média
MATHEMATICAL SCIENCES
A remark on soliton equation of mean curvature flow
Li Ma; Yang Yang
Department of Mathematics, Tsinghua University, 100084 Beijing, China
Correspondence Correspondence to Li Ma E-mail: lma@math.tsinghua.edu.cn
ABSTRACT
In this note, we consider self-similar immersions of the mean curvature flow and show that a graph solution of the soliton equation, provided it has bounded derivative, converges smoothly to a function which has some special properties (see Theorem 1.1).
Key words: soliton, Self-Similar, Mean curvature flow.
RESUMO
Nesta nota, consideramos imersões auto-semelhantes do fluxo de curvatura média, e mostramos que uma solução em forma de gráfico da equação de soliton converge diferencialmente, contanto que tenha derivada limitada, para um gráfico cuja função tem propriedades especiais (V. Teorema 1.1).
Palavras-chave: Soliton, Auto-similar, Fluxo de curvatura média.
1 INTRODUCTION
Let Mn+k be a Riemannian manifold of dimension n + k. Assume that Sn be a Riemannian manifold of dimension n without boundary. Let F : Sn ® Mn+k be an isometric immersion. Denote Ñ (respectively D) the covariant differentiation on S (on M). Let T S and N S be the tangent bundle and normal bundle of S in M respectively. We define the second fundamental form of the immersion S by
with
for tangential vector fields X, Y on S. We define the mean curvature vector field (in short, MCV) by
In recent years, many people are interested in studying the evolution of the immersion F : Sn® Mn+k along its Mean Curvature Flow (in short, just say MCF). The MCF is defined as follows. Given an one-parameter family of sub-manifolds St = Ft(S) with immersions Ft : S ® M . Let (t) be the MCV of St. Then our MCF is the equation/system
This flow has many very nice results if the codimension k = 1. See the work of Huisken 1993 for a survey in this regard. Since there is very few result about MCF in higher codimension, we will study it in the target when Mn+k = Rn+k, which is the standard Euclidian space.
In this short note, we will consider a family of self-similar graphic immersions F(·, t): n® n+k of the Mean Curvature Flow (MCF):
Write
and
By definition, we call the family Stself-similar if
In this case, we can reduce the MCF into an elliptic system. In the other word, we have the following parametric elliptic equation for the family St:
We will call this system as the soliton equation of the MCF. Note that this equation is usually obtained from the monotonicity formula of Huisken 1989 for blow-up. It is a hard and open problem to classify solutions of this equation.
Fix S = St. Assume that F(x) = (x, f(x)). Let
is the orthogonal projection onto NpS, where p Î S. Then the second fundamental form of S can be written as
Hence, we have the expression for the mean curvature vector of S in
n+k:
Our main result in this paper is the following
THEOREM 1.1. Let F(x) = (x, f(x)), x Î n be a graph solution to the soliton equation
Assume sup |Df(x)| < C0 < +¥. Then there exists a unique smooth function f¥ : n ® k such that
and
for any real number r ¹ 0, where
We remark that the proof of this result given below is very simple. But it is based on a nice observation. We just use the divergence theorem with a nice test function. In the next section, we recall the form of divergence theorem for convenient of the readers. In the last section we give a proof of our Theorem.
We point out that we may consider F¥(x) = (x, f¥(x)) obtained above as a tangential minimal cone along the research direction done by Simon 1983 (see also Ecker and Huisken 1989).
2 PRELIMINARY
Given a vector field X : S ® T M. Let XT and XN denote the projection of X onto T S and N S respectively. We define the divergence of X on S as
where (gij) = (gij)-1, and (gij) is the induced metric tensor written in local coordinates (xi) on S.
Note that, for any tangential vector field Y on S,
So
Hence
and by the Stokes formula on S, we have
and
where n is the exterior normal vector field to S on ¶S.
3 PROOF OF MAIN THEOREM
In the following, we take Mn+k = n+k as the standard Euclidean space. We assume that the assumption of our Theorem 1.1 is true in this section.
Define the vector field
where s Î to be determined.
Note that, and divSF = n. So
Locally, we may assume that S is a graph of the form (x, f(x)) Î BR(0) × k, where BR(0) is the ball of radius R centered at 0. Let SR = S Ç (BR(0) × k). By the divergence theorem we have (d):
Clearly we have that the left side of (d) is
By direct computation, the right side of (d) is
Hence, we have
Since |FT| < |F| < 1 + |F|, we have
Clearly we have
Combining these two inequalities together we get
Choosing s = n yields (*):
By our assumption we have that $C > 0 such that for F(x) = (x, f(x)) on S = n, we have
on S. Since
we know that
Hence
Therefore we get from (*) the key estimate (K):
We now go to the proof of our Theorem 1.1.
PROOF. Note that the mean curvature flow for the graph of f can be read as
The important fact about this equation is that it is invariant under the transformation
Compute
Here we have used the fact that
So
Hence
So, for x Î Sn-1, we have
Notice that, for m > l > 1,
The last inequality follows from the inequality (K). Therefore, we have the estimate (**):
This implies that (fl) is a Cauchy sequence in L2(Sn-1). Let f¥ be its unique limit. Since sup |Dfl| = sup |Df| < C0, the Arzela-Ascoli theorem tells us that (fl) is compact in Ca(Sn-1), "a Î (0, 1). Therefore
and
This finishes the proof of Theorem 1.1.
In the following, we pose a question about the stability of self-similar solutions of (MCF). Let f0 : n ® k be a smooth function with uniformly bounded (Lipschitz) gradient. Assume
Assume f : n × [0, ¥) ® k such that F(x, t) = (x, f(x, t)) is a solution of (MCF) with the initial data F(x, 0) = (x, f0(x)). We ask if there is a smooth mapping : n ® k such that (·, s) ® (·) uniformly on compact subsets of n as s ® ¥. Here is defined by
A related stability result is done by one of us in Ma 2003.
According to the remark of the referee, the codimension 1 case is settled in reference Stavrou 1998 with the trivial cone as only possible limit. A nice question now is that, can one give a condition that enforces the trivial cone in higher codimension? In Stavrou 1998, the stability for codimension 1 entire graphs with bounded gradient is treated - showing that they converge to asymptotically expanding solutions if they have a unique tangent cone at infinity. (This is of course not so relevant for the present paper, but may be related to our result in an interesting way).
ACKNOWLEDGMENTS
The work of Ma is partially supported by the key 973 project of China. We thank the referee for a useful suggestion.
Manuscript received on February 4, 2004; accepted for publication on February 8, 2004;
presented by MANFREDO DO CARMO
AMS Classification: 53C44, 53C42.
- ECKER K AND HUISKEN G. 1989. Mean curvature evolution of entire graphs. Ann Math 130: 453-471.
- HUISKEN G. 1989. Asymptotic behavior for singularities of the mean curvature flow. J Diff Geom 231: 285-299.
- HUISKEN G. 1993. Local and global behavior of hypersurfaces moving by mean curvature flow. Proc. of Symposia in Pure Math 54 (Part I): 175-191.
- MA L. 2003. B-sub-manifolds and their stability. math.DG/0304493.
- SIMON L. 1983. Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems, Ann Math 118: 525-571.
- STAVROU N. 1998. Selfsimilar solutions to the mean curvature flow. J Reine Angew Math 499: 189-198.
Publication Dates
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Publication in this collection
23 Aug 2004 -
Date of issue
Sept 2004
History
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Received
04 Feb 2004 -
Accepted
08 Feb 2004