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 degree angles, and expands homothetically under curve shortening flow. We also prove uniqueness of these networks.
@article{ASNSP_2007_5_6_4_511_0, author = {Schn\"urer, Oliver C. and Schulze, Felix}, title = {Self-similarly expanding networks to curve shortening flow}, journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze}, pages = {511--528}, publisher = {Scuola Normale Superiore, Pisa}, volume = {Ser. 5, 6}, number = {4}, year = {2007}, mrnumber = {2394409}, zbl = {1139.53031}, language = {en}, url = {http://www.numdam.org/item/ASNSP_2007_5_6_4_511_0/} }
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 SP - 511 EP - 528 VL - 6 IS - 4 PB - Scuola Normale Superiore, Pisa UR - http://www.numdam.org/item/ASNSP_2007_5_6_4_511_0/ LA - en ID - ASNSP_2007_5_6_4_511_0 ER -
%0 Journal Article %A Schnürer, Oliver C. %A Schulze, Felix %T Self-similarly expanding networks to curve shortening flow %J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze %D 2007 %P 511-528 %V 6 %N 4 %I Scuola Normale Superiore, Pisa %U http://www.numdam.org/item/ASNSP_2007_5_6_4_511_0/ %G en %F ASNSP_2007_5_6_4_511_0
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] “The Motion of a Surface by its Mean Curvature”, Mathematical Notes, Vol. 20, Princeton University Press, Princeton, N.J., 1978. | MR | Zbl
,[2] 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 | Zbl
and ,[3] Mean curvature evolution of entire graphs, Ann. of Math. 130 (1989), 453-471. | MR | Zbl
and ,[4] Interior estimates for hypersurfaces moving by mean curvature, Invent. Math. 105 (1991), 547-569. | MR | Zbl
and ,[5] The heat equation shrinking convex plane curves, J. Differential Geom. 23 (1986), 69-96. | MR | Zbl
and ,[6] The heat equation shrinks embedded plane curves to points, J. Differential Geom. 26 (1987), 285-314. | MR | Zbl
,[7] A distance comparison principle for evolving curves, Asian J. Math. 2 (1998), 127-133. | MR | Zbl
,[8] “Lectures on Mean Curvature Flow and Related Equations”, 1998, available from http://www.math.ethz.ch/ilmanen/
,[9] Motion by curvature of planar networks, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) 3 (2004), 235-324. | Numdam | MR | Zbl
, and ,[10] Self similar expanding solutions of the planar network flow, arXiv:0704.3113v1 [math.DG]. | MR | Zbl
and ,[11] Selfsimilar solutions to the mean curvature flow, J. Reine Angew. Math. 499 (1998), 189-198. | MR | Zbl
,[12] “Mengentheoretische Topologie”, Springer-Verlag, Berlin, 1973, Hochschultext. | MR | Zbl
,