Nonlinear potential theory and Ricci-pinched 3-manifolds

TL;DR

This paper proves that complete, noncompact 3-manifolds with Ricci-pinching are flat, using nonlinear potential theory under the assumption of superquadratic volume growth, providing an alternative proof of Hamilton's conjecture.

Contribution

It offers a new proof of Hamilton's Ricci-pinching conjecture for 3-manifolds using nonlinear potential theory with volume growth conditions.

Findings

Ricci-pinched 3-manifolds are flat under specified conditions

Alternative proof of Hamilton's conjecture

Volume growth condition is crucial for the proof

Abstract

In this paper, we focus on Hamilton's pinching conjecture formulated in Hamilton's paper "Three-manifolds with positive Ricci curvature". Let $(M, g)$ be a complete, connected, noncompact Riemannian $3$-manifold satisfying the Ricci-pinching condition. Then, it is flat. Here, we give an alternative proof, based on nonlinear potential theory, under the extra hypothesis of superquadratic volume growth.