A Note on Ricci-pinched three-manifolds

TL;DR

The paper tackles Hamilton's pinching conjecture in dimension three by delivering a direct potential-theoretic proof: a complete, non-compact Ricci-pinched $3$-manifold with $AVR>0$ is flat, and this extends to the broader class of manifolds with superquadratic volume growth, $α>4/3$, under the same curvature pinching. The authors construct a harmonic potential $u$ on $M\setminus\Omega$, define $w=-\log u$, and analyze level-sets via the monotone quantities $\mathscr{F}$ and $\mathscr{G}$, linking geometric constraints to volume growth through capacity estimates. A differential inequality $\mathscr{F}'(t)\le\max\{-2\mathscr{F}(t),\varepsilon(2\mathscr{F}(t)-8\pi)\}$ drives $\mathscr{F}(t)$ to zero, which contradicts superquadratic growth unless the manifold is flat; the argument extends to nonorientable manifolds via the orientable double cover and yields boundary-case corollaries. This work provides a flow-free, potential-theory-based alternative to Ricci-flow proofs of Hamilton's pinching conjecture and reinforces the link between curvature pinching and global geometry under volume-growth hypotheses.

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-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.

Theorems & Definitions (15)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3