The Curve Shortening Problem

BASIC RESULTS Short Time Existence Facts from Parabolic Theory Evolution of Geometric Quantities INVARIANT SOLUTIONS FOR THE CURVE SHORTENING FLOW Travelling Waves Spirals The Support Function of a Convex Curve Self-Similar Solutions THE CURVATURE-EIKONAL FLOW FOR CONVEX CURVES Blaschke Selection Theorem Preserving Convexity and Shrinking to a Point Gage-Hamilton Theorem The Contracting Case of the ACEF The Stationary case of the ACEF The Expanding Case of the ACEF THE CONVEX GENERALIZED CURVE SHORTENING FLOW Results from Brunn-Minkowski Theory The AGCSF for s in (1/3,1) The Affine Curve Shortening Flow Uniqueness of Self-Similar Solutions THE NON-CONVEX CURVE SHORTENING FLOW An Isoperimetric Ratio Limits of the Rescaled Flow Classification of Singularities A CLASS OF NON-CONVEX ANISOTROPIC FLOWS Decrease in Total Absolute Curvature Existence of a Limit Curve Shrinking to a Point A Whisker Lemma The Convexity Theorem EMBEDDED CLOSED GEODESICS ON SURFACES Basic Results The Limit Curve Shrinking to a Point Convergence to a Geodesic THE NON-CONVEX GENERALIZED CURVE SHORTENING FLOW Short Time Existence The Number of Convex Arcs The Limit Curve Removal of Interior Singularities The Almost Convexity Theorem BIBLIOGRAPHY