The inverse mean curvature flow perpendicular to the sphere

TL;DR

This paper studies the inverse mean curvature flow of convex hypersurfaces with boundary conditions, showing they converge to flat disks in a specific smoothness norm, advancing understanding of geometric flows with boundary constraints.

Contribution

It establishes convergence of inverse mean curvature flow for convex hypersurfaces with boundary perpendicular to the sphere, a new result in geometric flow theory.

Findings

Flow hypersurfaces converge to flat disks in $C^{1,eta}$ norm.

The convergence result applies to hypersurfaces with boundary perpendicular to the sphere.

Provides new insights into boundary behavior of inverse mean curvature flow.

Abstract

We consider the smooth inverse mean curvature flow of strictly convex hypersurfaces with boundary embedded in $\mathbb{R}^{n+1},$ which are perpendicular to the unit sphere from the inside. We prove that the flow hypersurfaces converge to the embedding of a flat disk in the norm of $C^{1,\beta},$ $\beta<1.$