Inverse Mean Curvature Flow of Rotationally Symmetric Hypersurfaces

TL;DR

This paper proves the long-term existence and convergence of inverse mean curvature flow for certain non-star-shaped, mean-convex hypersurfaces with symmetry, and explores applications to geometric inequalities and minimal surface theory.

Contribution

It extends inverse mean curvature flow results to non-star-shaped hypersurfaces with symmetry and applies these to Minkowski inequalities and minimal surface problems.

Findings

Flow exists for all time for specified hypersurfaces.

Flow converges to a round sphere homothetically.

Applications include extended Minkowski inequality and minimal surface embeddedness.

Abstract

We prove that the Inverse Mean Curvature Flow of a non-star-shaped, mean-convex embedded sphere in $\mathbb{R}^{n+1}$ with symmetry about an axis and sufficiently long, thick necks exists for all time and homothetically converges to a round sphere as $t \rightarrow \infty$. Our approach is based on a localized version of the parabolic maximum principle. We also present two applications of this result. The first is an extension of the Minkowski inequality to the corresponding non-star-shaped, mean-convex domains in $\mathbb{R}^{n+1}$. The second is a connection between IMCF and minimal surface theory. Based on previous work by Meeks and Yau and using foliations by IMCF, we establish embeddedness of the solution to Plateau's problem and a finiteness property of stable immersed minimal disks for certain Jordan curves in $\mathbb{R}^{3}$.