Inverse Mean Curvature Flow for Star-Shaped Hypersurfaces Evolving in a Cone

Abstract

For a given convex cone we consider hypersurfaces with boundary which are star-shaped with respect to the center of the cone and which meet the cone perpendicular. The evolution of those hypersurfaces inside the cone yields a nonlinear parabolic Neumann problem. We show that one can use the convexity of the cone to prove long time existence of this flow. Finally, we show that the hypersurfaces converge smoothly to a piece of the round sphere.

Capacity inequalities and rigidity of cornered/conical manifolds

“…Building on the ideas of Freire and Schwartz in their proof of mass-capacity inequalities [7], we obtain analogous results by considering now hypersurfaces with boundary evolving under inverse mean curvature flow, an approach introduced by Marquadt [21,22], who constructed solutions by rewriting the flow as an equation for the level set of a function whose advantage is to allow "jumps" in a natural way. For different approaches of this geometric flow, we refer the reader to [15,16,22].…”

“…In this section, we give a brief discussion of the inverse mean curvature flow for hypersurfaces that possesses boundary. Part of the proofs herein relies on modifications of the argument in [7] using the approach developed by Marquadt in [20,21,22].…”

Weak solutions of inverse mean curvature flow for hypersurfaces with boundary

“…Furthermore, after suitable rescaling the hypersurfaces converge to a piece of a round sphere [23]. For closed hypersurfaces this result goes back to Gerhardt [4] (see also Urbas [33]).…”

“…Brought to you by | University of California Authenticated Download Date | 6/11/15 9:18 PM sphere [23]. The aim of this work is to prove the existence of weak solutions of IMCF for hypersurfaces with boundary.…”

The inverse mean curvature flow perpendicular to the sphere

“…[16]. Here the IMCF of hypersurfaces with boundary was considered and the embedded flowing hypersurfaces were supposed to be perpendicular to a convex cone in R n+1 .…”