Convexity estimates for mean curvature flow with free boundary

Abstract

In this paper, we generalize White's regularity and structure theory for mean-convex mean curvature flow [34][35][36] 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 [8], Haslhofer-Hershkovits [14], and Volkmann [33]. Other important new ingredients are a Bernstein-type theorem and a sheeting theorem for low entropy free boundary flows in a halfslab, which allow us to rule out multiplicity 2 (half-)planes as possible tangent flows and, for mean convex domains, as possible limit flows.

The free-boundary Brakke flow

“…In the 90's Stahl [20] proved long-time existence of the smooth, compact, free-boundary mean curvature flow of hypersurfaces, in the sense that curvature blow-up must occur at a finite-time singularity. Some progress has been made analysing mean-convex singularities through smooth blow-ups: via a particular monotonicity formula Buckland [4] proved type-I singularities are modeled on generalized cylinders (with free-boundary in a plane), and recently the author [5] proved type-II singularities can be realized by translating solitons, via the Huisken-Sinestrari estimates. Many others have considered smooth freeboundary curvature flows, including [13], [19], [22], [14], [16].…”

Mean curvature flow with free boundary in embedded cylinders or cones and uniqueness results for minimal hypersurfaces

“…A similar angle approach has been employed by Lambert [32] in his work. Edelen's work is the first systematic treatment of Type II singularities [17]. Convexity estimates play a fundamental role in his work.…”

Mean curvature flow with free boundary – Type 2 singularities

“…Edelen's work is the first systematic treatment of Type 2 singularities . Convexity estimates play a fundamental role in his work.…”