Preprint

Inserted: 8 sep 2024
Last Updated: 10 sep 2024

Year: 2024
Links: arXiv, PDF

Abstract:

Let $(M, g)$ be a complete, connected, non-compact Riemannian $3$-manifold. Suppose that $(M,g)$ satisfies the Ricci-pinching condition $\mathrm{Ric}\geqslant\varepsilon\mathrm{R} g$ 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 by Huisken-Koerber 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.