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 Ric epsilon Rg for some epsilon > 0, where Ric and R are the Ricci tensor and the 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. This result was previously shown by Deruelle- Schulze-Simon [8] and Huisken-K & ouml;rber [14] and together with the contributions of Lott [17] and Lee-Topping [15], it led to a proof of the so-called Hamilton's pinching conjecture