2013/10/16 by Richard H. Bamler, Bamler, Richard H.
Mathematics · Medicine · #Advanced Neuroimaging Techniques and Applications #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Geometry and complex manifolds #math.AP #math.DG #math.GT
paper · pdf · doi:10.48550/arxiv.1310.4483
86 pages
arxiv created 2013/10/16 · openalex publication_date 2013/10/16 · arxiv updated 2013/10/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
In this paper we analyze the long-time behavior of 3 dimensional Ricci flows with surgery. Our main result is that if the surgeries are performed correctly, then only finitely many surgeries occur and after some time the curvature is bounded by C t-1. This result confirms a conjecture of Perelman. In the course of the proof, we also obtain a qualitative description of the geometry as t → ∞. This paper is the third part of a series. Previously, we had to impose a certain topological condition T2 to establish the finiteness of the surgeries and the curvature control. The objective of this paper is to remove this condition and to generalize the result to arbitrary closed 3-manifolds. This goal is achieved by a new area evolution estimate for minimal simplicial complexes, which is of independent interest.