2011/02/24 by Joseph Lauer, Lauer, Joseph, Joseph J. Lauer
Mathematics · #Advanced Topology and Set Theory #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Mathematical Dynamics and Fractals #Point processes and geometric inequalities #math.AP #math.DG
paper · pdf · doi:10.48550/arxiv.1102.5110
23 pages, 5 figures
openalex publication_date 2011/02/24 · arxiv created 2011/12/01 · arxiv updated 2011/12/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 γn is a sequence of smooth curves converging uniformly to J, then the lengths \mathscrL(γnt), where γnt denotes the result of applying curve shortening flow to γn 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.