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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Angenent S.: The zeroset of a solution of a parabolic equation. Journal für die reine und angewandte Mathematik, 390, 79–96 (1988)
Chen Y.G., Giga Y., Goto S.: Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations. Journal of Differential Geometry, 33(3), 749–786 (1991)
Ecker K., Huisken G.: Mean curvature evolution of entire graphs. Annals of Mathematics, 130, 453–471 (1989)
Ecker K., Huisken G.: Interior estimates for hypersurfaces moving by their mean curvature. Invented Mathematics, 105, 547–569 (1991)
L.C. Evans. Partial Differential Equations. American Mathematical Society , Providence, ISBN 0821807722 (1998)
Evans L.C., Spruck J.: Motion of level sets by mean curvature. . I. Journal of Differential Geometry, 33(3), 635–681 (1991)
Gage M., Hamilton R.: The heat equation shrinking convex plane curves. Journal of Differential Geometry, 23, 69–96 (1986)
Grayson M.: The heat equation shrinks embedded plane curves to round points. Journal of Differential Geometry, 26, 285–314 (1987)
Huisken G.: Shrinking convex spheres by their mean curvature. Journal of Differential Geometry, 20, 237–266 (1984)
T. Ilmanen. The level set flow on a manifold. Differential Geometry: partial differential equations on manifolds (Los Angeles, CA, 1990). In: Proceedings of Symposia in Pure Mathematics, vol 54. (1993), pp. 193–204
T. Ilmanen. Elliptic regularization and partial regularity for motion by mean curvature. Memoirs of the American Mathematical Society, (520)108 (1994), x+90
A. Polden. Evolving Curves. Honours Thesis. Australian National University, Canberra (1991)
White B.: The size of the singular set in mean curvature flow of mean-convex sets. Journal of the American Mathematical Society, 13(3), 665–695 (2000)
White B.: Nature of singularities in mean curvature flow of mean-convex sets. Journal of the American Mathematical Society, 16(1), 123–138 (2003)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Lauer, J. A New Length Estimate for Curve Shortening Flow and Low Regularity Initial Data. Geom. Funct. Anal. 23, 1934–1961 (2013). https://doi.org/10.1007/s00039-013-0248-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00039-013-0248-1