Bounding dimension of ambient space by density for mean curvature flow

Abstract

For an ancient solution of the mean curvature flow, we show that each time slice M t 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 M t that evolves under mean curvature flow for all negative time t. IntroductionThis paper deals with ancient solutions of mean curvature flow. An ancient solution is a family (M t ) 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 M t 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 (M t ) t∈(a,b) of n-dimensional submanifolds of R n+k moves by mean curvature if there exist immersions x t = x(·, t) : M n −→ R n+k of an n-dimensional manifold M n with images M t = x t (M n ) satisfying the evolution equationHere H(p, t) denotes the mean curvature vector of M t at x(p, t) for (p, t) ∈ M n × (a, b). The space-time track of the family (M t ) is the setIn particular, a minimal n-dimensional submanifold M of R n+k 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.

Complexity of parabolic systems

“…) for all (x, t) with x ∈ M t , t < 0 . (0.4) Motivated by [CM1]- [CM5], similar spaces were considered in Calle's thesis [Ca1], [Ca2].…”

On Ecker's local integral quantity at infinity for ancient mean curvature flows

“…In particular, when M is ancient, we say that M is well-defined (see [1]) in R N × (−∞, 0) if each M t has no boundary in R N and has locally finite mass…”

“…Using the relation (1.10), the assumption on entropy in Theorem 1.5 can be replaced by the assumption on Ecker's integral quantity (see also Calle [1]).…”

Liouville properties

“…In view of [CM4], [CM5] and [CM6] for harmonic functions, it is natural to seek dimension bounds for these spaces on manifolds. This was initiated by Calle in 2006 in her thesis, [Ca1], [Ca2]; cf. [LZ] for some recent results, including a parabolic generalization of [CM4].…”