Mean Convex Mean Curvature Flow with Free Boundary

Abstract

In this paper, we generalize White's regularity and structure theory for meanconvex 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 tripleapproximation scheme, which combines ideas from Edelen [9], Haslhofer-Hershkovits [16], and Volkmann [43]. Other important new ingredients are a Bernsteintype 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.

Mean curvature flow with generic initial data

“…It seems likely that our arguments here could be extended to n ≥ 7, but some care would need to be taken when constructing the flows in Proposition 7.3. See [Whi15,HH18,EHIZ19] for techniques related to this issue. 8.2.…”

“…We expect that the dimensional restriction in Theorems 9.1 and 9.2 can be removed (cf. [Whi15,HH18,EHIZ19]). We note that when Σ has sufficiently small F -area, Theorems 9.1 and 9.2 hold in all dimensions.…”

Mean curvature flow with generic initial data

“…We expect that the dimensional restriction in Theorems 9.1 and 9.2 can be removed (cf. [56,62,111]). We note that when has sufficiently small F -area, Theorems 9.1 and 9.2 hold in all dimensions.…”