Mass Drop and Multiplicity in Mean Curvature Flow

TL;DR

This paper investigates conditions under which Brakke flows, a type of mean curvature flow, do not experience mass drop, establishing links with multiplicity conjectures and providing cases where equality in the Brakke inequality holds.

Contribution

It proves that flows without mass drop satisfy the multiplicity one conjecture and identifies conditions ensuring no mass drop and equality in the Brakke inequality.

Findings

No mass drop for flows with mean convex neighborhoods of singularities.

Generic flows have no mass drop until higher multiplicity tangent flows appear.

Flows with three-convex blow-up types satisfy the Brakke inequality with equality.

Abstract

Brakke flow is defined with a variational inequality, which means it may have discontinuous mass over time, i.e. have mass drop. It has long been conjectured that the Brakke flow associated to a nonfattening level set flow has no mass drop and achieves equality in the Brakke inequality. Under natural assumptions, we show that a flow has no mass drop if and only if it satisfies the multiplicity one conjecture $\mathcal{H}^n$-a.e. One application is that there is no mass drop for level set flows with mean convex neighborhoods of singularities, and a generic flow has no mass drop until there is a higher multiplicity planar tangent flow. Also, if a nonfattening flow has no higher multiplicity planes as limit flows, then each limit flow has no mass drop. We upgrade these results to equality in the Brakke inequality for certain important cases. We show that nonfattening flows with three-convex blow-up type are Brakke flows with equality. This includes flows with generic singularities in dimension three and flows with mean convex neighborhoods of singularities in dimension four.