Isoperimetric comparison techniques for low dimension curvature flows

Abstract

Two low dimension curvature flows are studied: the Ricci flow on surfaces and the curve shortening flow of embedded closed curves in the plane. The main theorems proven are that the corresponding normalised flows have solutions existing for all time and which converge to a minimising configuration, namely one with constant curvature. The theorems follow from comparison theorems for isoperimetric quantities. For the Ricci flow, the isoperimetric profile is used. For the curve shortening flow, two different isoperimetric quantities are used leading to two separate proofs of the main theorem. The first quantity is the isoperimetric profile of the interior of the curve whilst the second is a chord/arc ratio. In all cases, the basic approach is to compare the isoperimetric quantity with that of a suitable model solution which has isoperimetric quantity initially below the given solution and which converges to the constant curvature solution. An application of the maximum principle then shows that any arbitrary solution is bounded by the model solution for all time. This in turn leads directly to strong control over the curvature and isoperimetric constant of arbitrary solutions which provides analytic control through the Sobolev constant. The main theorems then follow from fairly standard arguments. Although the main theorems were previously known, the comparison theorems described here are relatively elementary and lead much more directly to the main theorems than previous proofs. -- provided by Candidate.