Strong noncollapsing and uniform Sobolev inequalities for Ricci flow with surgeries

Qi S. Zhang

Vol. 239 (2009), No. 1, 179–200

DOI: 10.2140/pjm.2009.239.179

Abstract

We prove a uniform Sobolev inequality for Ricci flow that is independent of the number of surgeries. As an application, under fewer assumptions, we derive a noncollapsing result stronger than Perelman’s κ-noncollapsing result with surgery. The proof is shorter and seems more accessible. The result also improves some earlier ones where the Sobolev inequality depended on the number of surgeries.

Keywords

strong noncollapsing, Sobolev imbedding, Ricci flow, surgery

Mathematical Subject Classification 2000

Primary: 53C44

Milestones

Received: 29 July 2008

Revised: 15 September 2008

Accepted: 17 September 2008

Published: 1 January 2009