Positively curved shrinking Ricci solitons are compact

2015/04/29 by Ovidiu Munteanu, Jiaping Wang, Munteanu, Ovidiu +1 · 5 citations
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.1504.07898

openalex publication_date 2015/04/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that a shrinking Ricci soliton with positive sectional curvature must be compact. This extends a result of Perelman in dimension three and improves a result of Naber in dimension four, respectively.

Citations

• Ricci solitons: the equation point of view
• Ricci flow with surgery on three-manifolds

Cited by

• Gradient Shrinking Ricci Solitons and Modified Sectional Curvature
• On Shrinking Ricci solitons with positive isotropic curvature in higher dimensions
• Geometry and Topology of Gradient Shrinking Sasaki-Ricci Solitons
• A compactness theorem for four-dimensional shrinking gradient Ricci solitons
• Structure at infinity for shrinking Ricci solitons

Related

• Sharp upper diameter bounds for compact shrinking Ricci solitons
• Shrinking Ricci solitons with positive isotropic curvature
• On Shrinking Ricci solitons with positive isotropic curvature in higher dimensions
• Compact Gradient Shrinking Ricci Solitons with Positive Curvature Operator
• Expanding Ricci solitons with pinched Ricci curvature