A Note on Ricci-pinched three-manifolds

TL;DR

This paper provides an alternative proof, using potential theory, that Ricci-pinched three-manifolds with Euclidean volume growth are flat, contributing to the understanding of Hamilton's pinching conjecture.

Contribution

It offers a new potential-theoretic proof of flatness for Ricci-pinched three-manifolds with Euclidean volume growth, complementing existing results.

Findings

Ricci-pinched three-manifolds with Euclidean volume growth are flat

Alternative proof based on potential theory

Supports Hamilton's pinching conjecture

Abstract

Let $(M, g)$ be a complete, connected, non-compact Riemannian $3$-manifold. Suppose that $(M,g)$ satisfies the Ricci--pinching condition $\mathrm{Ric}\geq\varepsilon\mathrm{R} g$ for some $\varepsilon>0$, where $\mathrm{Ric}$ and $\mathrm{R}$ are the Ricci tensor and scalar curvature, respectively. In this short note, we give an alternative proof based on potential theory of the fact that if $(M,g)$ has Euclidean volume growth, then it is flat. Deruelle-Schulze-Simon and Huisken-K\"{o}rber have already shown this result and together with the contributions by Lott and Lee-Topping led to a proof of the so-called Hamilton's pinching conjecture.