https://neckpinch.com/authors/xiaodong-cao/New arXiv papers by Xiaodong Cao, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2509.20669arxiv:2509.20669Thu, 25 Sep 2025 00:00:00 GMTXiaodong Cao, Ernani Ribeiro, Hosea Wondo — We investigate four-dimensional gradient shrinking Ricci solitons with positive modified sectional curvature. Our first main result shows that if the norm of the self-dual Weyl tensor and the scalar curvature satisfy a certain sharp pinching condition (closely related, in a precise sense, to that of Kähler metric), then the soliton is necessarily locally Kähler. We further obtain a characterization theorem and a weighted integral gap result for compact gradient shrinking Ricci solitons with bounded modified sectional curvature. In addition, we establish a Hitchin-Thorpe type inequality for compact four-dimensional Ricci solitons, providing new topological constraints on such manifolds.Ricci solitonsCurvature pinching4-manifoldsKähler geometryScalar curvature https://arxiv.org/abs/2409.13123arxiv:2409.13123Thu, 19 Sep 2024 00:00:00 GMTXiaodong Cao, Hung Tran — [Dedicated to Richard S. Hamilton on forty years of Ricci flow] Gradient Ricci solitons have garnered significant attention both as self-similar solutions and singularity models of the Ricci flow. This survey article starts with a list of examples; it also provides some geometric aspects of gradient Ricci solitons, including various asymptotic behaviors; finally, it discusses some recent results on classification and rigidity. In particular, this survey focuses on dimension four.Ricci flowRicci solitonsSingularities4-manifoldsStability & uniqueness https://arxiv.org/abs/2212.05267arxiv:2212.05267Sat, 10 Dec 2022 00:00:00 GMTXiaodong Cao, Ernani Ribeiro, Hung Tran — In this article, we investigate four-dimensional gradient shrinking Ricci solitons close to a Kähler model. The first theorem could be considered as a rigidity result for the Kähler-Ricci soliton structure on S²× ℝ² (in the sense of Remark 1). Moreover, we show that if the quotient of norm of the self-dual Weyl tensor and scalar curvature is close to that on a Kähler metric in a specific sense, then the gradient Ricci soliton must be either half-conformally flat or locally Kähler.Ricci solitonsKähler–Ricci solitons4-manifoldsKähler geometryScalar curvatureStability & uniqueness