Some Foundational Results for Free Boundary Brakke Flows

Abstract

In this paper, we establish some geometric and analytic foundations for free boundary Brakke flows. Specifically, we (i) introduce unit-regular and cyclic free boundary flows and show that they are preserved under reflections and weak limits, (ii) prove that the support of free boundary Brakke flows satisfies an avoidance principle, and (iii) introduce free boundary inner and outer flows and prove the existence of matching free boundary Brakke flows. These results serve as general tools to analyze free boundary flows through singularities, and in particular will be applied in forthcoming work with Haslhofer, where we address the mean-convex neighborhood conjecture and uniqueness conjecture for free boundary flows through (half) cylindrical singularities.