Inverse mean curvature flows in warped product manifolds

TL;DR

This paper investigates inverse mean curvature flows of starshaped, mean convex hypersurfaces in warped product manifolds, establishing long-term existence, preservation of geometric properties, and conditions for extending Euclidean asymptotic behaviors.

Contribution

It provides new conditions ensuring the long-term existence and geometric preservation of inverse mean curvature flows in warped product manifolds, extending Euclidean results.

Findings

Flows exist for all time under specified conditions.

Mean curvature remains positively bounded independently of initial curvature.

Conditions identified for extending Euclidean asymptotic behavior to warped products.

Abstract

We study inverse mean curvature flows of starshaped, mean convex hypersurfaces in warped product manifolds with a positive warping factor $\varphi(r)$. If $\varphi'(r)>0$ and $\varphi''(r)\geq 0$, we show that these flows exist for all times, remain starshaped and mean convex. Plus the positivity of $\varphi''(r)$ and a curvature condition we obtain a lower positive bound of mean curvature along these flows independent of the initial mean curvature. We also give a sufficient condition to extend the asymptotic behavior of these flows in Euclidean spaces into some more general warped product manifolds.