A Note on Ricci-pinched three-manifolds

Let $(M,g)$ be a complete, connected, non-compact Riemannian $3$-manifold. Suppose that $(M,g)$ satisfies the Ricci–pinching condition $\mathrm{R}\mathrm{i}\mathrm{c}\ge \epsilon \mathrm{R}g$ for some $\epsilon >0$, where $\mathrm{R}\mathrm{i}\mathrm{c}$ 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-Körber 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.