Ricci flow of almost non-negatively curved three manifolds

✍ Scribed by Simon, Miles

• Book ID: 120033898
• Publisher: Walter de Gruyter GmbH & Co. KG
• Year: 2009
• Tongue: English
• Weight: 328 KB
• Volume: 2009
• Category: Article
• ISSN: 0075-4102
• DOI: 10.1515/CRELLE.2009.038

✦ Synopsis

In this paper we study the evolution of almost non-negatively curved (possibly singular) three dimensional metric spaces by Ricci flow. The non-negatively curved metric spaces which we consider arise as limits of smooth Riemannian manifolds ðM i ; i gÞ, i A N, whose Ricci curvature is bigger than À1=i, and whose diameter is less than d 0 (independent of i) and whose volume is bigger than v 0 > 0 (independent of i). We show for such spaces, that a solution to Ricci flow exists for a short time t A ð0; TÞ, that the solution is smooth for t > 0, and has Ricci À gðtÞ Á f 0 and Riem À gðtÞ Á e c=t for t A ð0; TÞ (for some constant c ¼ cðv 0 ; d 0 ; nÞ). This allows us to classify the topological type and the di¤erential structure of the limit manifold (in view of the theorem of Hamilton [10] on closed three manifolds with non-negative Ricci curvature).