Mean Convex Mean Curvature Flow with Free Boundary

Abstract

In this paper, we generalize White's regularity and structure theory for mean-convex mean curvature flow [45, 46, 48] to the setting with free boundary. A major new challenge in the free boundary setting is to derive an a priori bound for the ratio between the norm of the second fundamental form and the mean curvature. We establish such a bound via the maximum principle for a triple-approximation scheme, which combines ideas from Edelen [9], Haslhofer-Hershkovits [16], and Volkmann [43]. Other important new ingredients are a Bernstein-type theorem and a sheeting theorem for low-entropy free boundary flows in a half-slab, which allow us to rule out multiplicity 2 (half-)planes as possible tangent flows and, for mean-convex domains, as possible limit flows. © 2021 Wiley Periodicals LLC.