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. 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 $\gamma_n$ is a sequence of smooth curves converging uniformly to $J$, then the lengths $\mathscr{L}({\gamma_n}_t)$, where ${\gamma_n}_t$ denotes the result of applying curve shortening flow to $\gamma_n$ 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 curve shortening flow has a unique solution if the initial data is a finite length Jordan curve.