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

27 pages, Lemma 3.4 corrected, asymptotic estimates added

References in corpus (5)

• Inverse curvature flows in Riemannian warped products
• Inverse Mean Curvature Flow and the Stability of the Positive Mass Theorem and Riemannian Penrose Inequality Under $L^2$ Metric Convergence
• Stability of the Positive Mass Theorem and Riemannian Penrose Inequality for Asymptotically Hyperbolic Manifolds Foliated by Inverse Mean Curvature Flow
• Homothetic solitons for the inverse mean curvature flow
• ODE Maximum Principle at Infinity and Non-Compact Solutions of IMCF in Hyperbolic Space

Cited by in corpus (2)

• Inverse curvature flows in Riemannian warped products
• Sobolev stability of the PMT and RPI using IMCF