Paper · Sep 25, 2025
Gradient Shrinking Ricci Solitons and Modified Sectional Curvature
As on arXiv
Abstract
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.
Comments: To appear in Advances in Mathematics. Comments are welcomed
Its neighbourhood
Connected papers
The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2021, plus the canon.
-
Rigidity of four-dimensional Kähler-Ricci solitons
Topics 4-manifolds, Scalar curvature, Kähler geometry · Same authors Xiaodong Cao, Ernani Ribeiro · Shared terms “four dimensional”, “dual Weyl”, “investigate four”
-
On Euler characteristic and Hitchin-Thorpe inequality for four-dimensional compact Ricci solitons
Topics 4-manifolds, Ricci solitons · Same author Ernani Ribeiro · Shared terms “Hitchin Thorpe”, “compact gradient”, “four dimensional”
-
Topological Obstructions to the Existence of Compact Shrinking Ricci Solitons in Dimension Four
Topics 4-manifolds, Kähler geometry, Ricci solitons · Shared terms “shrinking Ricci”, “gradient shrinking”, “Hitchin Thorpe”
-
Geometry and Analysis of Gradient Ricci Solitons in Dimension Four
Topics 4-manifolds, Ricci solitons · Same author Xiaodong Cao
-
A note on four dimensional Shrinking Gradient Ricci Solitons with Constant Scalar Curvature
Topics 4-manifolds, Scalar curvature, Ricci solitons · Shared terms “four dimensional”, “gradient shrinking”, “shrinking Ricci”
-
A note on Rigidity of Shrinking Gradient Ricci Solitons with Constant Scalar Curvature
Topics 4-manifolds, Scalar curvature, Ricci solitons · Shared terms “gradient shrinking”, “shrinking Ricci”, “sectional curvature”
-
Curvature pinching of asymptotically conical gradient expanding Ricci solitons
Topics Curvature pinching, 4-manifolds, Scalar curvature · Shared terms “four dimensional”, “sectional curvature”, “compact gradient”
-
Sharp Weitzenböck and PIC2 Estimates from Sectional-Scalar Curvature Pinching
Topics Curvature pinching, Scalar curvature · Shared terms “sectional curvature”, “pinching condition”
Read around it
Background reading
The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.
In the guidebook
- 6A.9Reaction–Diffusion: Fronts, Patterns and Pinching
Critical patch size, travelling fronts and their speeds, bistable waves and the Allen–Cahn equation, Turing patterns, and the curvature of a Ricci flow as a reaction–diffusion system.
- 11A.7Maximum Principles under Ricci Flow
Scalar and tensor maximum principles, and preserved curvature conditions.
- 11A.10Hamilton–Ivey Pinching
Why high curvature in dimension 3 is almost positive.