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

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## 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.

## Keywords and phrases

Curve shortening flow Level set flow## Mathematics Subject Classification (2000)

53C44## Preview

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