The Curve Shortening ProblemAlthough research in curve shortening flow has been very active for nearly 20 years, the results of those efforts have remained scattered throughout the literature. For the first time, The Curve Shortening Problem collects and illuminates those results in a comprehensive, rigorous, and self-contained account of the fundamental results. The a |
Contents
| 1 | |
2 Invariant Solutions for the Curve Shortening Flow | 27 |
3 The CurvatureEikonal Flow for Convex Curves | 45 |
4 The Convex Generalized Curve Shortening Flow | 93 |
5 The Nonconvex Curve Shortening Flow | 121 |
6 A Class of Nonconvex Anisotropic Flows | 143 |
7 Embedded Closed Geodesics on Surfaces | 179 |
8 The Nonconvex Generalized Curve Shortening Flow | 203 |
| 247 | |
| 254 | |
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Common terms and phrases
6-whisker a-axis affine curvature assume blow-up c(to C*-norm Cauchy problem Chapter concave arc connected convex arc convex curve convex function convex sets convex\concave arcs curve shortening flow defined Denote depends disjoint disk embedded closed curve endpoints evolving arc F is parabolic fact finite foliation follows graph Harnack inequality Hausdorff metric Hence Hölder inequality inflection points initial curve interior intersection invariant isoperimetric ratio Lemma length limit curve line segment Lipschitz continuous maximum principle mean curvature Minkowski inequality normal angle normalized flow P(to parabolic equation parametrization positive constants positive lower bound resp satisfies shrinks singularities smooth solution of 1.2 strictly increasing strong maximum principle Sturm oscillation theorem subarc subconverges support function Suppose symmetric t t w tangent angle tends to zero total absolute curvature total curvature uniformly bounded uniformly convex uniformly parabolic unique vertical line vertical point whisker lemma