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

Oliver C. Schnürer, Abderrahim Azouani, Marc Georgi, Juliette Hell, Nihar Jangle, Amos Koeller, Tobias Marxen, Sandra Ritthaler, Mariel Sáez, Felix Schulze, Brian Smith

Research output: Contribution to journalArticle

14 Scopus citations

Abstract

We consider convex symmetric lens-shaped networks in ℝ2 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.

Original languageEnglish (US)
Pages (from-to)2265-2294
Number of pages30
JournalTransactions of the American Mathematical Society
Volume363
Issue number5
DOIs
StatePublished - May 1 2011
Externally publishedYes

Keywords

  • Mean curvature flow
  • Networks
  • Triple junctions

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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    Schnürer, O. C., Azouani, A., Georgi, M., Hell, J., Jangle, N., Koeller, A., Marxen, T., Ritthaler, S., Sáez, M., Schulze, F., & Smith, B. (2011). Evolution of convex lens-shaped networks under the curve shortening flow. Transactions of the American Mathematical Society, 363(5), 2265-2294. https://doi.org/10.1090/S0002-9947-2010-04820-2