A New Length Estimate for Curve Shortening Flow and Low Regularity Initial Data

Abstract

In this paper we introduce a geometric quantity, the r-multiplicity, that controls the length of a smooth curve as it evolves by curve shortening flow (CSF). The length estimates we obtain are used to prove results about the level set flow in the plane. If K is locally-connected, connected and compact, then the level set flow of K either vanishes instantly, fattens instantly or instantly becomes a smooth closed curve. If the compact set in question is a Jordan curve J, then the proof proceeds by using the r-multiplicity to show that if γ n is a sequence of smooth curves converging uniformly to J, then the lengths $${\fancyscript{L}({\gamma_n}_t)}$$ , where γ n t denotes the result of applying CSF to γ n for time t, are uniformly bounded for each t > 0. Once the level set flow has been shown to be smooth we prove that the Cauchy problem for CSF has a unique solution if the initial data is a finite length Jordan curve.