The Boundary Term in Huisken's Monotonicity Formula and the Entropy of Translators

TL;DR

This paper investigates how boundary terms affect entropy in mean curvature flow, providing bounds for the entropy of translators based on boundary geometry, thus advancing understanding of geometric flow behavior with boundaries.

Contribution

It introduces a natural control of boundary terms in Huisken's monotonicity formula, relating translator entropy to boundary entropy and cone density under mild conditions.

Findings

Boundary term controlled by geometric boundary properties

Entropy of translators bounded by boundary entropy and cone density

Provides new insights into boundary effects in mean curvature flow

Abstract

For a manifold-with-boundary moving by mean curvature flow, the entropy at a later time is bounded by the entropy at an earlier time plus a boundary term. This paper controls the boundary term in a geometrically natural way. In particular, it shows (under mild hypotheses)that the entropy of a compact translator is less than or equal to the entropy of the boundary plus the maximal cone density of the boundary.