On inverse mean curvature flow in Schwarzschild space and Kottler space

TL;DR

This paper investigates the inverse mean curvature flow in Schwarzschild and Kottler-Schwarzschild spaces, demonstrating convergence to large spheres and establishing global existence for certain initial hypersurfaces, extending previous results.

Contribution

It provides new analysis of inverse mean curvature flow in Schwarzschild and Kottler spaces, including convergence behavior and global existence results for star-shaped hypersurfaces.

Findings

Flow hypersurfaces converge exponentially to large spheres in Schwarzschild space.

Established a lower bound for mean curvature independent of initial conditions.

Proved global existence and regularity of the flow for star-shaped, weakly mean convex hypersurfaces.

Abstract

In this paper, we first study the behavior of inverse mean curvature flow in Schwarzschild manifold. We show that if the initial hypersurface $\Sigma$ is strictly mean convex and star-shaped, then the flow hypersurface $\Sigma_t$ converges to a large coordinate sphere as $t\rightarrow \infty$ exponentially. We also describe an application of this convergence result. In the second part of this paper, we will analyse the inverse mean curvature flow in Kottler-Schwarzchild manifold. By deriving a lower bound for the mean curvature on the flow hypersurface independently of the initial mean curvature, we can use an approximation argument to show the global existence and regularity of the smooth inverse mean curvature flow for star-shaped and weakly mean convex initial hypersurface, which generalizes Huisken-Ilmanen's result [18].