Hyperbolic inverse mean curvature flow

TL;DR

This paper establishes the short-time existence and analyzes the behavior of hyperbolic inverse mean curvature flow for mean convex, star-shaped hypersurfaces, including expansion and convergence properties in the plane.

Contribution

It proves the short-time existence of hyperbolic inverse mean curvature flow for certain initial conditions and explores its geometric evolution and convergence in the plane.

Findings

Existence of solutions for mean convex, star-shaped hypersurfaces.

Examples of hyperbolic evolution equations for geometric quantities.

Results on expansion and convergence of the flow in the plane.

Abstract

In this paper, we prove the short-time existence of hyperbolic inverse (mean) curvature flow (with or without the specified forcing term) under the assumption that the initial compact smooth hypersurface of $\mathbb{R}^{n+1}$ ($n\geqslant2$) is mean convex and star-shaped. Several interesting examples and some hyperbolic evolution equations for geometric quantities of the evolving hypersurfaces have been shown. Besides, under different assumptions for the initial velocity, we can get the expansion and the convergence results of a hyperbolic inverse mean curvature flow in the plane $\mathbb{R}^2$, whose evolving curves move normally.