Bounding Dimension of Ambient Space by Density for Mean Curvature Flow

For an ancient solution of the mean curvature flow, we show that each time slice Mt is contained in an affine subspace with dimension bounded in terms of the density and the dimension of the evolving submanifold. Recall that an ancient solution is a family Mt that evolves under mean curvature flow for all negative time t. 0. Introduction This paper deals with ancient solutions of mean curvature flow. An ancient solution is a family (Mt) of n-dimensional submanifold of R n+k that moves by mean curvature flow for all negative time t (or in general, for all times t < T for some fixed T ). We prove that each Mt is contained in an affine subspace of bounded dimension. The bound on the dimension depends only on a bound on the density and on the dimension of the evolving manifold. A family (Mt)t∈(a,b) of n-dimensional submanifolds of R n+k moves by mean curvature if there exist immersions xt = x(·, t) : M n −→ R of an n-dimensional manifold M with images Mt = xt(M ) satisfying the evolution equation ∂x ∂t = ~ H. (1) Here ~ H(p, t) denotes the mean curvature vector of Mt at x(p, t) for (p, t) ∈ M n × (a, b). The space-time track of the family (Mt) is the set M = ⋃ t∈(a,b) Mt × {t} ⊂ R n+k ×R, sometimes simply written as M = {(Mt, t), t ∈ (a, b)}. In particular, a minimal n-dimensional submanifold M of R is a stationary solution of the evolution equation (1), because in M we have ~ H = 0. Mean curvature flow can be defined not only for smooth manifolds, but also for more general objects. In particular, in [B] Brakke defines mean curvature flow for integral varifolds. A varifold is a measure-theoretic generalization of a manifold that can have singularities. Often a smooth solution of (1) develops singularities in finite time, and after that it becomes a varifold solution (also called a weak solution). Most of what we state in this paper for smooth solutions of mean curvature flow is also true for weak solutions. Date: February 1, 2008. 1 2 MARIA CALLE For a minimal n-dimensional submanifold the density at a point x0 is defined by Θ(M,x0) = limr→0+ H(M∩Br) rn . For a solution of (1), instead of the area H(M ∩ Br) we consider an integral quantity that we denote by A(M ∩ Er), defined by Ecker in [E1] (see section 1 of this paper for a precise definition). This quantity plays the role of the area H(M ∩ Br), in particular the density for a mean curvature flow solution M at a point (x0, t0) ∈ M is defined by Θ(M, x0, t0) = limr→0+ A(M∩Er) rn . Philosophically, one can think of the ratio A(M∩Er) rn as a measure for how close the spacetime track M is to R. For M = R, this ratio is constantly equal to 1. As we will see, the main result in this paper goes along this line: we prove that when this ratio is bounded uniformly for all r, the manifolds Mt are contained in affine subspaces of dimension possibly smaller than the dimension of the ambient space R. For a solution of mean curvature flow, this ratio is nondecreasing in r (see section 1), and its limit when r goes to 0 is called the density. For smooth solutions the density is always 1. On the other hand, unit density gives regularity: a weak solution with unit density almost everywhere is smooth (see [B]). Our main result is the following: Theorem 1. Let M be an ancient solution of MCF in R such that ∀t ∈ (−∞, 0), Mt has no boundary in R and has finite mass H(Mt ∩ B2 √ −2nt) < ∞. If M satisfies: 1 ≤ A(M∩ Er) rn ≤ VM < ∞ ∀r > 0, then for each t ∈ (−∞, 0), Mt is contained in some affine subspace with dim ≤ (n+ 1) n n− 1 2 VM. The main point of this bound on the dimension is that it only depends on n, but not on k. We have stated the theorem for smooth solutions of (1), but in fact we will see that it also holds for varifold solutions of mean curvature flow. As mentioned above, this is important since smooth solutions very often develop singularities. In the theorem we ask that the manifolds Mt, t ∈ (−∞, 0) have no boundary in R n+k and have finite mass H(Mt∩B2 √ −2nt) < ∞. An ancient solution M with this property is called well-defined in R. Throughout the paper we assume that M is a well-defined ancient solution. As seen in [E2], this guarantees that all integral quantities considered in the paper are finite. The organization of the paper is as follows: in the first section, we recall some facts about mean curvature flow, in particular a mean value formula proved by Ecker in [E2] that will be essential for our proof. In this section, we also develop a little the concept of weak solutions of mean curvature flow, and give some examples of smooth ancient solutions. In section 2, we give the statement of a second theorem, from which our main theorem is a corollary. Also, we give an idea of the steps of the proof. Many of the ideas for this proof are inspired BOUNDING DIMENSION OF AMBIENT SPACE BY DENSITY FOR MEAN CURVATURE FLOW 3 by the paper [CM2] of Colding and Minicozzi. In section 3, we give the statement and proof of several lemmas, necessary for the proof of theorem 2. Finally, in section 4 we give the proof of theorem 2. I would like to thank my advisor Tobias Colding for his help and for suggesting this problem. 1. Preliminaires In this section we introduce some definitions and formulas about mean curvature flow, give an idea of how Brakke solutions of mean curvature flow are defined, and give some examples of ancient solutions. A minimal n-dimensional submanifold M on R is a stationary solution of the evolution equation (1). For such a manifold M , we have the following monotonicity formula (see [CM1]): d dr ( H(M ∩Br) rn )