Relative expander entropy in the presence of a two-sided obstacle and applications

TL;DR

This paper introduces a new relative entropy concept for hypersurfaces constrained by obstacles, develops its variational theory, and extends monotonicity formulas in mean curvature flow, enhancing understanding of geometric flows with obstacles.

Contribution

It defines a relative entropy for hypersurfaces between two self-expanders, establishes its existence, and develops a variational framework for obstacle problems in mean curvature flow.

Findings

Existence of relative entropy for hypersurfaces between self-expanders

Development of variational theory for the relative entropy functional

A version of the forward monotonicity formula for mean curvature flow

Abstract

We study a notion of relative entropy motivated by self-expanders of mean curvature flow. In particular, we obtain the existence of this quantity for arbitrary hypersurfaces trapped between two disjoint self-expanders asymptotic to the same cone. This allows us to begin to develop the variational theory for the relative entropy functional for the associated obstacle problem. We also obtain a version of the forward monotonicity formula for mean curvature flow proposed by Ilmanen.