Curvature pinching of asymptotically conical gradient expanding Ricci solitons
Huai-Dong Cao, Junming Xie
TL;DR
This work extends Hamilton-Ivey-type curvature pinching from shrinking and steady Ricci solitons to non-compact asymptotically conical gradient expanding Ricci solitons. By adapting barrier-maximum techniques and a general differential-inequality framework, the authors establish several pinching results: in 3D, AC expanders with cones of positive scalar curvature have positive sectional curvature; in higher dimensions with vanishing Weyl, they obtain positive curvature operator; in 4D they prove results under (half) PIC and UPIC, including nonnegativity and positivity of the curvature operator and rotational symmetry under non-flat cone asymptotics. A general lemma (5.1) provides a unified tool to transfer barrier-type inequalities into global nonnegativity, which the authors apply to derive the UPIC theorem and additional pinching statements. The results deepen the parallel between expanding solitons and the classical shrinking/steady theory, with implications for uniqueness, rigidity, and potential classifications of AC expanders under curvature conditions.
Abstract
Since the well-known work of Hamilton [62] and Ivey [64], the Hamilton-Ivey curvature pinching and its generalizations have become a signature feature of gradient shrinking and steady Ricci solitons, and more generally, of ancient solutions to the Ricci flow. However, analogous results for gradient expanding Ricci solitons have remained elusive. This is largely due to the fact that the proofs of existing curvature pinching estimates crucially rely on shrinking and steady solitons being ancient, a property not shared by gradient Ricci expanders. In this paper, we investigate curvature pinching phenomena in non-compact asymptotically conical gradient expanding Ricci solitons and establish several Hamilton-Ivey type curvature pinching estimates. These results are parallel to those known for shrinking and steady Ricci solitons. In particular, we prove a three-dimensional Hamilton-Ivey type curvature pinching theorem: any three-dimensional non-compact gradient Ricci expander, which is asymptotic to a cone with positive scalar curvature, must have positive sectional curvature. As an application, we combine our result with that of Deruelle [51] to establish a uniqueness theorem for three-dimensional asymptotically conical expanders with positive scalar curvature. Furthermore, we formulate a general proof method and apply it to obtain analogues of several additional known generalized Hamilton-Ivey type curvature pinching results for ancient solutions. Among these is a curvature pinching estimate for four-dimensional asymptotically conical Ricci expanders with uniformly positive isotropic curvature, analogous to a result for four-dimensional gradient steady solitons due to Brendle [7].