Self-similarly expanding networks to curve shortening flow

Oliver C. Schnürer; Felix Schulze

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2007)

  • Volume: 6, Issue: 4, page 511-528
  • ISSN: 0391-173X

Abstract

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We consider a network in the Euclidean plane that consists of three distinct half-lines with common start points. From that network as initial condition, there exists a network that consists of three curves that all start at one point, where they form 120 degree angles, and expands homothetically under curve shortening flow. We also prove uniqueness of these networks.

How to cite

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Schnürer, Oliver C., and Schulze, Felix. "Self-similarly expanding networks to curve shortening flow." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 6.4 (2007): 511-528. <http://eudml.org/doc/272269>.

@article{Schnürer2007,
abstract = {We consider a network in the Euclidean plane that consists of three distinct half-lines with common start points. From that network as initial condition, there exists a network that consists of three curves that all start at one point, where they form $120$ degree angles, and expands homothetically under curve shortening flow. We also prove uniqueness of these networks.},
author = {Schnürer, Oliver C., Schulze, Felix},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
language = {eng},
number = {4},
pages = {511-528},
publisher = {Scuola Normale Superiore, Pisa},
title = {Self-similarly expanding networks to curve shortening flow},
url = {http://eudml.org/doc/272269},
volume = {6},
year = {2007},
}

TY - JOUR
AU - Schnürer, Oliver C.
AU - Schulze, Felix
TI - Self-similarly expanding networks to curve shortening flow
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2007
PB - Scuola Normale Superiore, Pisa
VL - 6
IS - 4
SP - 511
EP - 528
AB - We consider a network in the Euclidean plane that consists of three distinct half-lines with common start points. From that network as initial condition, there exists a network that consists of three curves that all start at one point, where they form $120$ degree angles, and expands homothetically under curve shortening flow. We also prove uniqueness of these networks.
LA - eng
UR - http://eudml.org/doc/272269
ER -

References

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  8. [8] T. Ilmanen, “Lectures on Mean Curvature Flow and Related Equations”, 1998, available from http://www.math.ethz.ch/ ∼ ilmanen/ 
  9. [9] C. Mantegazza, M. Novaga and V. M. Tortorelli, Motion by curvature of planar networks, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) 3 (2004), 235–324. Zbl1170.53313MR2075985
  10. [10] R. Mazzeo and M. Sáez, Self similar expanding solutions of the planar network flow, arXiv:0704.3113v1 [math.DG]. Zbl1280.53062MR3060453
  11. [11] N. Stavrou, Selfsimilar solutions to the mean curvature flow, J. Reine Angew. Math.499 (1998), 189–198. Zbl0895.53039MR1631112
  12. [12] B. von Querenburg, “Mengentheoretische Topologie”, Springer-Verlag, Berlin, 1973, Hochschultext. Zbl0431.54001MR467641

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