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MINIMALLY INVASIVE SURGERY FOR RICCI FLOW SINGULARITIES
 

Summary: MINIMALLY INVASIVE SURGERY
FOR RICCI FLOW SINGULARITIES
SIGURD B. ANGENENT, M. CRISTINA CAPUTO, AND DAN KNOPF
Abstract. In this paper, we construct smooth forward Ricci flow evolutions
of singular initial metrics resulting from rotationally symmetric neckpinches
on Sn+1, without performing an intervening surgery. In the restrictive context
of rotational symmetry, this construction gives evidence in favor of Perelman's
hope for a "canonically defined Ricci flow through singularities".
Contents
1. Introduction 2
2. Recovering from a neckpinch singularity 6
2.1. The initial metric and its regularizations 6
2.2. Evolution equations for u(r, t) and v(r, t) = u(r, t)2
8
2.3. The shape of the singular initial data in the r variables 9
3. Possible complete smooth solutions 9
3.1. Lower barriers for v 10
3.2. Yet another maximum principle 11
4. Formal matched asymptotics 13
4.1. The outer region (r 1) 13

  

Source: Angenent, Sigurd - Department of Mathematics, University of Wisconsin at Madison

 

Collections: Mathematics