Topological Change in Mean Convex Mean Curvature Flow

TL;DR

This paper investigates the topological changes in mean convex mean curvature flow, establishing conditions under which certain homotopy groups can vanish and constructing examples with specific singularities.

Contribution

It proves that homotopy groups can only vanish through specific shrinking singularities and constructs examples with prescribed singularities in mean curvature flow.

Findings

Homotopy groups can only die via shrinking S^k x R^(n-k) singularities.

Existence of regions with prescribed S^m x R^(n-m) singularities.

Characterization of topological change mechanisms in mean convex mean curvature flow.

Abstract

Consider the mean curvature flow of an (n+1)-dimensional, compact, mean convex region in Euclidean space (or, if n<7, in a Riemannian manifold). We prove that elements of the m-th homotopy group of the complementary region can die only if there is a shrinking S^k x R^(n-k) singularity for some k less than or equal to m. We also prove that for each m from 1 to n, there is a nonempty open set of compact, mean convex regions K in R^(n+1) with smooth boundary for which the resulting mean curvature flow has a shrinking S^m x R^(n-m) singularity.