Evolution of convex lens-shaped networks under the curve shortening flow
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- by Oliver C. Schnürer, Abderrahim Azouani, Marc Georgi, Juliette Hell, Nihar Jangle, Amos Koeller, Tobias Marxen, Sandra Ritthaler, Mariel Sáez, Felix Schulze and Brian Smith PDF
- Trans. Amer. Math. Soc. 363 (2011), 2265-2294 Request permission
Abstract:
We consider convex symmetric lens-shaped networks in $\mathbb {R}^2$ that evolve under the curve shortening flow. We show that the enclosed convex domain shrinks to a point in finite time. Furthermore, after appropriate rescaling the evolving networks converge to a self-similarly shrinking network, which we prove to be unique in an appropriate class. We also include a classification result for some self-similarly shrinking networks.References
- U. Abresch and J. Langer, The normalized curve shortening flow and homothetic solutions, J. Differential Geom. 23 (1986), no. 2, 175–196. MR 845704, DOI 10.4310/jdg/1214440025
- Ben Andrews, Classification of limiting shapes for isotropic curve flows, J. Amer. Math. Soc. 16 (2003), no. 2, 443–459. MR 1949167, DOI 10.1090/S0894-0347-02-00415-0
- Xinfu Chen and Jong-Shenq Guo, Self-similar solutions of a 2-D multiple-phase curvature flow, Phys. D 229 (2007), no. 1, 22–34. MR 2340176, DOI 10.1016/j.physd.2007.03.009
- Julie Clutterbuck, Parabolic equations with continuous initial data, arXiv:math.AP/0504455.
- Tobias H. Colding and William P. Minicozzi II, Sharp estimates for mean curvature flow of graphs, J. Reine Angew. Math. 574 (2004), 187–195. MR 2099114, DOI 10.1515/crll.2004.069
- Klaus Ecker and Gerhard Huisken, Mean curvature evolution of entire graphs, Ann. of Math. (2) 130 (1989), no. 3, 453–471. MR 1025164, DOI 10.2307/1971452
- Klaus Ecker and Gerhard Huisken, Interior estimates for hypersurfaces moving by mean curvature, Invent. Math. 105 (1991), no. 3, 547–569. MR 1117150, DOI 10.1007/BF01232278
- M. Gage and R. S. Hamilton, The heat equation shrinking convex plane curves, J. Differential Geom. 23 (1986), no. 1, 69–96. MR 840401, DOI 10.4310/jdg/1214439902
- Matthew A. Grayson, The heat equation shrinks embedded plane curves to round points, J. Differential Geom. 26 (1987), no. 2, 285–314. MR 906392
- Jörg Hättenschweiler, Mean curvature flow of networks with triple junctions in the plane, Diplomarbeit, Zürich: ETH Zürich, Department of Mathematics, 46 pp., 2007.
- Gerhard Huisken, Asymptotic behavior for singularities of the mean curvature flow, J. Differential Geom. 31 (1990), no. 1, 285–299. MR 1030675
- Gerhard Huisken, A distance comparison principle for evolving curves, Asian J. Math. 2 (1998), no. 1, 127–133. MR 1656553, DOI 10.4310/AJM.1998.v2.n1.a2
- Carlo Mantegazza, Matteo Novaga, and Vincenzo Maria Tortorelli, Motion by curvature of planar networks, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 3 (2004), no. 2, 235–324. MR 2075985
- W. W. Mullins, Two-dimensional motion of idealized grain boundaries, J. Appl. Phys. 27 (1956), 900–904. MR 78836, DOI 10.1063/1.1722511
- John von Neumann, Discussion to ‘Grain shapes and other metallurgical applications of topology’ by C. S. Smith, Metal Interfaces, Amer. Soc. for Metals, Cleveland, Ohio, R. M. Brick (Editor), 1952, pp. 108–110.
Additional Information
- Oliver C. Schnürer
- Affiliation: Department of Mathematics, Freie Universität Berlin, Arnimallee 3, 14195 Berlin, Germany
- Address at time of publication: Fachbereich Mathematik und Statistik, Universität Konstanz, 78457 Konstanz, Germany
- Email: Oliver.Schnuerer@math.fu-berlin.de, Oliver.Schnuerer@uni-konstanz.de
- Abderrahim Azouani
- Affiliation: Department of Mathematics, Freie Universität Berlin, Arnimallee 3, 14195 Berlin, Germany
- Email: azouani@math.fu-berlin.de
- Marc Georgi
- Affiliation: Department of Mathematics, Freie Universität Berlin, Arnimallee 3, 14195 Berlin, Germany
- Email: georgi@math.fu-berlin.de
- Juliette Hell
- Affiliation: Department of Mathematics, Freie Universität Berlin, Arnimallee 3, 14195 Berlin, Germany
- Email: blanca@math.fu-berlin.de
- Nihar Jangle
- Affiliation: Department of Mathematics, Freie Universität Berlin, Arnimallee 3, 14195 Berlin, Germany
- Email: jangle@math.fu-berlin.de
- Amos Koeller
- Affiliation: Mathematisches Institut der Eberhard-Karls-Universität Tübingen, Auf der Morgenstelle 10, D-72076 Tübingen, Germany
- Email: akoeller@everest.mathematik.uni-tuebingen.de
- Tobias Marxen
- Affiliation: Department of Mathematics, Freie Universität Berlin, Arnimallee 3, 14195 Berlin, Germany
- Email: marxen@math.fu-berlin.de
- Sandra Ritthaler
- Affiliation: Department of Mathematics, Freie Universität Berlin, Arnimallee 3, 14195 Berlin, Germany
- Email: ritthale@math.fu-berlin.de
- Mariel Sáez
- Affiliation: Max Planck Institute for Gravitational Physics (Albert Einstein Institute), Wissenschaftspark Golm, Am Mühlenberg 1, 14476 Golm, Germany
- Address at time of publication: Edificio Rolando Chuaqui, Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Avda. Vicuña Mackenna 4860, Macul, Santiago, Chile
- Email: mariel@aei.mpg.de, mariel@mat.puc.cl
- Felix Schulze
- Affiliation: Department of Mathematics, Freie Universität Berlin, Arnimallee 3, 14195 Berlin, Germany
- Email: Felix.Schulze@math.fu-berlin.de
- Brian Smith
- Affiliation: Department of Mathematics, Freie Universität Berlin, Arnimallee 3, 14195 Berlin, Germany
- Email: bsmith@math.fu-berlin.de
- Received by editor(s): April 29, 2008
- Published electronically: December 2, 2010
- Additional Notes: Several authors are members of and were supported by SFB 647/B3 “Raum – Zeit – Materie: Singularity structure, long-time behavior and dynamics of solutions of non-linear evolution equations”. The ninth author was supported by the grant “Proyecto Fondecyt de Iniciación 11070025”
- © Copyright 2010 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 363 (2011), 2265-2294
- MSC (2010): Primary 53C44, 35B40
- DOI: https://doi.org/10.1090/S0002-9947-2010-04820-2
- MathSciNet review: 2763716