Regularity of inverse mean curvature flow in asymptotically hyperbolic manifolds with dimension $3$

TL;DR

This paper studies the inverse mean curvature flow in three-dimensional asymptotically hyperbolic manifolds, showing regularity of the flow and properties of Hawking mass limits, with implications for geometric analysis in such spaces.

Contribution

It establishes the regularity of weak inverse mean curvature flow in asymptotically hyperbolic manifolds and analyzes the Hawking mass behavior, extending results beyond the asymptotically flat case.

Findings

Slices become star-shaped after long time

Regularity of the weak solution is proven

Hawking mass limit exceeds or equals total mass

Abstract

By making use of the nice behavior of Hawking masses of slices of a weak solution of inverse mean curvature flow in three dimensional asymptotically hyperbolic manifolds, we are able to show that each slice of the flow is star-shaped after a long time, and then we get the regularity of the weak solution of inverse mean curvature flow in asymptotically hyperbolic manifolds. As an application, we prove that the limit of Hawking mass of the slices of a weak solution of inverse mean curvature flow with any connected $C^2$-smooth surface as initial data in asymptotically ADS-Schwarzschild manifolds with positive mass is bigger than or equal to the total mass, which is completely different from the situation in asymptotically flat case.