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

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.

arXiv comment11 pages. The new version allows the boundary to include vertical portions. Newest version (Aug 5, 2023) corrects a typo in Theorem 16