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Geometric and Functional Analysis

, Volume 23, Issue 6, pp 1934–1961 | Cite as

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

  • Joseph LauerEmail author
Article

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 

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Copyright information

© Springer Basel 2013

Authors and Affiliations

  1. 1.Department of MathematicsYale UniversityNew HavenUSA

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