Inverse Mean Curvature Flow of Rotationally Symmetric Hypersurfaces

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 $t\to \infty$. 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 ${\mathbb{R}}^3$.

40 pages, 7 figures. This version includes more exposition on rotationally symmetric immersions, some additional applications, and a simpler example of a non-star-shaped admissible initial surface

References in corpus (3)

• Lectures on Minimal Surface Theory
• Inverse Mean Curvature Flow over Non-Star-Shaped Surfaces
• Embedded Plateau Problem

Cited by in corpus (1)

• The Limit of the Inverse Mean Curvature Flow on a Torus