A new length estimate for curve shortening flow and low regularity initial data

TL;DR

This paper introduces the $r$-multiplicity to estimate the length of curves under flow, leading to new results on the behavior of low-regularity initial data in curve shortening flow and level set flow.

Contribution

The paper develops the $r$-multiplicity as a geometric tool to control curve length and proves new regularity and uniqueness results for low-regularity initial data in curve shortening flow.

Findings

Length estimates via $r$-multiplicity control curve evolution.

Level set flow of certain sets either vanishes, fattens, or becomes smooth instantly.

Uniqueness of solutions for initial Jordan curves with finite length.

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.