2013/04/24 by Spyros Alexakis, Alexakis, Spyros, Dezhong Chen +3 · 1 citation
Mathematics · Physics and Astronomy · #35K20 #35K59 #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.1304.6449
openalex publication_date 2013/04/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We introduce certain spherically symmetric singular Ricci solitons and study their stability under the Ricci flow from a dynamical PDE point of view. The solitons in question exist for all dimensions n+1≥ 3, and all have a point singularity where the curvature blows up; their evolution under the Ricci flow is in sharp contrast to the evolution of their smooth counterparts. In particular, the family of diffeomorphisms associated with the Ricci flow "pushes away" from the singularity causing the evolving soliton to open up immediately becoming an incomplete (but non-singular) metric. In the second part of this paper we study the local-in time stability of this dynamical evolution, under spherically symmetric perturbations of the initial soliton metric. We prove a local well-posedness result for the Ricci flow near the singular initial data, which in particular implies that the "opening up" of the singularity persists for the perturbations also.