Rigidity of asymptotically conical shrinking gradient Ricci solitons

2013/07/17 by Brett Kotschwar, Lu Wang, Kotschwar, Brett +1 · 4 citations
Mathematics · Physics and Astronomy · #35K40 #53C44 #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.1307.4746

openalex publication_date 2013/07/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that if two gradient Ricci solitons are asymptotic along some end of each to the same regular cone, then the soliton metrics must be isometric on some neighborhoods of infinity of these ends. Our theorem imposes no restrictions on the behavior of the metrics off of the ends in question and in particular does not require their geodesic completeness. As an application, we prove that the only complete connected gradient shrinking Ricci soliton asymptotic to a rotationally symmetric cone is the Gaussian soliton on Rⁿ.

Citations

• Backwards uniqueness of the Ricci flow
• Unique continuation for the vacuum Einstein equations
• Sharp logarithmic Sobolev inequalities on gradient solitons and applications
• Uniqueness of Self-similar Shrinkers with Asymptotically Conical Ends
• Ricci solitons: the equation point of view
• Lower Bounds on Ricci Curvature and the Almost Rigidity of Warped Products
• On Bach-flat gradient shrinking Ricci solitons
• Ricci flow with surgery on three-manifolds

Cited by

• Curvature estimates for steady and expanding solitons in higher dimensions
• On curvature estimates for four-dimensional gradient Ricci solitons
• Gradient Shrinking Ricci Solitons and Modified Sectional Curvature
• Backwards uniqueness for Mean curvature flow with asymptotically conical singularities

Related

• Some rigidity results on shrinking gradient Ricci soliton
• Uniqueness of asymptotically cylindrical gradient shrinking Ricci solitons
• A uniqueness theorem for asymptotically cylindrical shrinking Ricci solitons
• Isometries of asymptotically conical shrinking Ricci solitons
• Rigidity of positively curved Steady gradient Ricci solitons on orbifolds