Is a complete Riemannian manifold with positively pinched Ricci curvature compact
Lei Ni
TL;DR
The paper investigates whether a complete Riemannian manifold with positively pinched Ricci curvature must be compact, offering a self-contained alternate proof of Hamilton's convex-hypersurface result through the lens of quasi-conformal maps. Central to the approach is the entropy-conformal metric $$\tilde{g}_{ij}=\frac{1}{f^2\log^2 f}g_{ij}$$ on a noncompact convex hypersurface $M^n\subset\mathbb{R}^{n+1}$, and the quasi-conformal nature of the Gauss map $\nu:M\to\mathbb{S}^n$ under the pinching condition $II_{ij}\ge \epsilon H g_{ij}$, which allows the application of Ni98 (Mich) to preclude noncompactness when $\lambda_1(N)>0$ and $V(o,r)/r^n\to0$. The paper then provides a detailed construction of the Ni98 argument—via a cut-off function, pullback to $N$, and analysis of singular values $\kappa_i$—to obtain a contradiction, establishing the desired compactness. It further extends the framework to generalizations involving Sobolev inequalities and finite-volume targets, offering testing cases for gradient shrinking, steady, and expanding solitons (including Kähler and non-Kähler settings) and linking to broader rigidity phenomena in nonnegative Ricci curvature. Overall, the work contributes a streamlined, flow-free route to compactness under Ricci-pinching and illuminates the stability of these rigidity results under various geometric-analytic hypotheses.
Abstract
A result of R. Hamilton asserts that any convex hypersurface in an Euclidian space with pinched second fundamental form must be compact. Partly inspired by this result, twenty years ago, in \cite{Ancient}, Remark 3.1 on page 650, the author formulated a problem asking if a complete Riemannian manifold with positively pinched Ricci curvature must be compact. There are several recent progresses, which are all rigidity results concerning the flat metric except the special case for the steady solitons. In this note we provide a detailed alternate proof of Hamilton's result, in view of the recent proof via the mean curvature flow requiring additional assumptions and that the original argument by Hamilton does lack of complete details. The proof uses a result of the author in 1998 concerning quasi-conformal maps. The proof here allows a generalization as well. We dedicate this article to commemorate R. Hamilton, the creator of the Ricci flow, who also made fundamental contributions to many other geometric flows.