Singularities of mean convex level set flow in general ambient manifolds

TL;DR

This paper establishes new estimates for mean convex level set flow in Riemannian manifolds, enabling the extension of structure theory to more general ambient spaces.

Contribution

It provides the first exponential-in-time control of mean curvature and second fundamental form ratios in general ambient manifolds, advancing the understanding of mean convex flows.

Findings

Exponential control of mean curvature in general manifolds

Control of second fundamental form to mean curvature ratio

Extension of structure theory for mean convex flows

Abstract

We prove two new estimates for the level set flow of mean convex domains in Riemannian manifolds. Our estimates give control - exponential in time - for the infimum of the mean curvature, and the ratio between the norm of the second fundamental form and the mean curvature. In particular, the estimates remove a stumbling block that has been left after the work of White and Haslhofer-Kleiner, and thus allow us to extend the structure theory for mean convex level set flow to general ambient manifolds of arbitrary dimension.