Geometric PDE Seminar

If a solution (M,g(t)) of Ricci flow develops a local singularity at a finite time T , then there is a proper subset S of M on which the curvature becomes infinite as time approaches T . Existing approaches to Ricci-flow-with-surgery, due to Hamilton and Perelman, require one to modify the solution in a small neighborhood of S by gluing in a highly curved but nonetheless nonsingular solution. This must be done with careful regard to various surgery parameters in order to preserve critical a priori estimates. In case the local singularity is a rotationally-symmetric neckpinch (in any dimension n > 2 ), we can now restart Ricci flow directly from the singular limit g(T) , without performing an intervening surgery or making ad hoc choices. We show that any complete smooth forward evolution from g(T) has a unique asymptotic profile as it emerges from the singularity, which we describe. (This is joint work with Sigurd Angenent and Cristina Caputo.)