Inverse mean curvature flow and Ricci-pinched three-manifolds

Abstract

Abstract Let ( M , g ) (M,g) be a noncompact, connected, complete Riemannian three-manifold with nonnegative Ricci curvature satisfying Ric ≥ ε ⁢ tr ⁡ ( Ric ) ⁢ g \mathrm{Ric}\geq\varepsilon\operatorname{tr}(\mathrm{Ric})g for some ε > 0 \varepsilon>0 . In this note, we give a new proof based on inverse mean curvature flow that ( M , g ) (M,g) is either flat or has non-Euclidean volume growth. In conjunction with the work of J. Lott [On 3-manifolds with pointwise pinched nonnegative Ricci curvature, Math. Ann. 388 (2024), 3, 2787–2806] and of M.-C. Lee and P. Topping [Three-manifolds with non-negatively pinched Ricci curvature, preprint (2022), https://arxiv.org/abs/2204.00504], this gives an alternative proof of a conjecture of R. Hamilton recently proven by A. Deruelle, F. Schulze, and M. Simon [Initial stability estimates for Ricci flow and three dimensional Ricci-pinched manifolds, preprint (2022), https://arxiv.org/abs/2203.15313] using Ricci flow.