An Intersection Principle for Mean Curvature Flow

TL;DR

This paper generalizes the avoidance principle for mean curvature flows, showing that intersection dimensions do not increase over time and extending these results to various weak flow solutions.

Contribution

It introduces a new intersection dimension monotonicity principle for mean curvature flows and extends it to Brakke and level set flows under certain conditions.

Findings

Hausdorff dimension of intersections is non-increasing over time

Self-intersection dimensions are non-increasing over time

Monotonicity fails for some weak solutions

Abstract

The avoidance principle says that mean curvature flows of hypersurfaces remain disjoint if they are disjoint at the initial time. We prove several generalizations of the avoidance principle that allow for intersections of hypersurfaces. First, we prove that the Hausdorff dimension of the intersection of two mean curvature flows is non-increasing over time, and we find precise information on how the dimension changes. We then show that the self-intersection of an immersed mean curvature flow has non-increasing dimension over time. Next, we extend the intersection dimension monotonicity to Brakke flows and level set flows which satisfy a localizability condition, and we provide examples showing that the monotonicity fails for general weak solutions. We find a localization result for level set flows with finitely many singularities, and as a consequence, we obtain a fattening criterion for these flows which depends on the behavior of intersections with smooth flows.