The constant angle problem for mean curvature flow inside rotational tori

Abstract

We flow a hypersurface in Euclidean space by mean curvature flow with a Neumann boundary condition, where the boundary manifold is any torus of revolution. If we impose the conditions that the initial manifold is compatible and does not contain the rotational vector field in its tangent space, then mean curvature flow exists for all time and converges to a flat cross-section as t → ∞.

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

“…Working in the class of graphical hypersurfaces is a viable strategy, so long as the graph condition can be preserved [47,49,48,31]. In each of these works, global results were enabled by symmetry of the initial data and/or of the boundary.…”

The inverse mean curvature flow perpendicular to the sphere

“…Other choices of boundary manifolds for a graphical mean curvature flow have shown convergence of the flow to flat disks, see for example [10,12], as well as [9] for a levelset approach.…”

Mean curvature flow with free boundary – Type 2 singularities

“…Working in the class of graphical hypersurfaces is a viable strategy, so long as the graph condition can be preserved. In each of , global results were enabled by symmetry of the initial data and/or of the boundary. Without such symmetries, recent work indicates that graphicality is not in general preserved (even in the case where is a standard round sphere).…”