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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Evolution of convex lens-shaped networks under the curve shortening flow


Authors: Oliver C. Schnürer, Abderrahim Azouani, Marc Georgi, Juliette Hell, Nihar Jangle, Amos Koeller, Tobias Marxen, Sandra Ritthaler, Mariel Sáez, Felix Schulze and Brian Smith
Journal: Trans. Amer. Math. Soc. 363 (2011), 2265-2294
MSC (2010): Primary 53C44, 35B40
Published electronically: December 2, 2010
MathSciNet review: 2763716
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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.


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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

DOI: http://dx.doi.org/10.1090/S0002-9947-2010-04820-2
PII: S 0002-9947(2010)04820-2
Keywords: Mean curvature flow, networks, triple junctions.
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”
Article copyright: © Copyright 2010 American Mathematical Society