Topological Change 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.
Keywords
Cite
@article{arxiv.1107.4644,
title = {Topological Change in Mean Convex Mean Curvature Flow},
author = {Brian White},
journal= {arXiv preprint arXiv:1107.4644},
year = {2013}
}
Comments
19 pages. This version includes a new section proving that certain kinds of mean curvature flow singularities persist under arbitrary small perturbations of the initial surface. Newest update (Oct 2013) fixes some bibliographic references