Long Time Existence of IMCF on Metrics Conformal to Warped Product Manifolds

TL;DR

This paper proves the long-time existence of inverse mean curvature flow on manifolds conformal to warped products, leveraging conformal vector fields to connect flow stability with geometric inequalities.

Contribution

It establishes long-time existence of IMCF on conformal warped product manifolds by analyzing conformal vector fields and their relation to the flow.

Findings

Long-time existence of IMCF is proven under specific conformal conditions.

Connections are made between IMCF stability and geometric inequalities like PMT and RPI.

Method provides tools for analyzing geometric flows on conformally warped manifolds.

Abstract

In this paper we study Inverse Mean Curvature Flow (IMCF) on manifolds that are conformal to a warped product manifold. To this end, we show how the gradient conformal vector field in warped product manifolds is related to the conformal vector field on the conformal metric and use this to gain control of the flow in order to show long time existence. Connections are made to recent results of the author on stability of the positive mass theorem (PMT) and the Riemannian Penrose inequality (RPI) where long time existence of IMCF is an important assumption.