Inverse mean curvature flow and Ricci-pinched three-manifolds

TL;DR

This paper provides a new proof using inverse mean curvature flow to classify complete non-compact three-manifolds with non-negative Ricci curvature, showing they are either flat or have non-Euclidean volume growth, supporting a conjecture related to Ricci flow.

Contribution

It introduces a novel proof technique based on inverse mean curvature flow for classifying Ricci-pinched three-manifolds, offering an alternative to Ricci flow methods.

Findings

Manifolds are either flat or have non-Euclidean volume growth.

Provides an alternative proof of a Ricci flow conjecture.

Connects inverse mean curvature flow with Ricci curvature classification.

Abstract

Let $(M,g)$ be a complete, connected, non-compact Riemannian three-manifold with non-negative Ricci curvature satisfying $Ric\geq\varepsilon\,\operatorname{tr}(Ric)\,g$ for some $\varepsilon>0$. In this note, we give a new proof based on inverse mean curvature flow that $(M,g)$ is either flat or has non-Euclidean volume growth. In conjunction with results of J. Lott and of M.-C. Lee and P. Topping, this gives an alternative proof of a conjecture of R. Hamilton recently proven by A. Deruelle, F. Schulze, and M. Simon using Ricci flow.