Self-similarly expanding networks to curve shortening flow
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 6 (2007) no. 4, pp. 511-528.

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.

Classification : 53C44,  35Q51,  74K30,  74N20
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     title = {Self-similarly expanding networks to curve shortening flow},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
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Schnürer, Oliver C.; Schulze, Felix. Self-similarly expanding networks to curve shortening flow. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 6 (2007) no. 4, pp. 511-528. http://www.numdam.org/item/ASNSP_2007_5_6_4_511_0/

[1] K. A. Brakke, “The Motion of a Surface by its Mean Curvature”, Mathematical Notes, Vol. 20, Princeton University Press, Princeton, N.J., 1978. | MR 485012 | Zbl 0386.53047

[2] L. Bronsard and F. Reitich, On three-phase boundary motion and the singular limit of a vector-valued Ginzburg-Landau equation, Arch. Ration. Mech. Anal. 124 (1993), 355-379. | MR 1240580 | Zbl 0785.76085

[3] K. Ecker and G. Huisken, Mean curvature evolution of entire graphs, Ann. of Math. 130 (1989), 453-471. | MR 1025164 | Zbl 0696.53036

[4] K. Ecker and G. Huisken, Interior estimates for hypersurfaces moving by mean curvature, Invent. Math. 105 (1991), 547-569. | MR 1117150 | Zbl 0707.53008

[5] M. E. Gage and R. S. Hamilton, The heat equation shrinking convex plane curves, J. Differential Geom. 23 (1986), 69-96. | MR 840401 | Zbl 0621.53001

[6] M. A. Grayson, The heat equation shrinks embedded plane curves to points, J. Differential Geom. 26 (1987), 285-314. | MR 906392 | Zbl 0667.53001

[7] G. Huisken, A distance comparison principle for evolving curves, Asian J. Math. 2 (1998), 127-133. | MR 1656553 | Zbl 0931.53032

[8] T. Ilmanen, “Lectures on Mean Curvature Flow and Related Equations”, 1998, available from http://www.math.ethz.ch/ilmanen/

[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. | Numdam | MR 2075985 | Zbl 1170.53313

[10] R. Mazzeo and M. Sáez, Self similar expanding solutions of the planar network flow, arXiv:0704.3113v1 [math.DG]. | MR 3060453 | Zbl 1280.53062

[11] N. Stavrou, Selfsimilar solutions to the mean curvature flow, J. Reine Angew. Math. 499 (1998), 189-198. | MR 1631112 | Zbl 0895.53039

[12] B. Von Querenburg, “Mengentheoretische Topologie”, Springer-Verlag, Berlin, 1973, Hochschultext. | MR 467641 | Zbl 0431.54001