Soliton solutions to the curve shortening flow on the sphere

dos Reis, Hiuri; Tenenblat, Keti

doi:10.1090/proc/14607

Soliton solutions to the curve shortening flow on the sphere
HTML articles powered by AMS MathViewer

by Hiuri Fellipe Santos dos Reis and Keti Tenenblat PDF

Proc. Amer. Math. Soc. 147 (2019), 4955-4967 Request permission

Abstract:

It is shown that a curve on the unit sphere is a soliton solution to the curve shortening flow if and only if its geodesic curvature is proportional to the inner product between its tangent vector and a fixed vector of $\mathbb {R}^3$. Using this characterization, we describe the geometry of such a curve on the sphere, we study its qualitative behavior, and we prove the convergence of the curve to the equator orthogonal to the fixed vector.

References

U. Abresch and J. Langer, The normalized curve shortening flow and homothetic solutions, J. Differential Geom. 23 (1986), no. 2, 175–196. MR 845704, DOI 10.4310/jdg/1214440025
Sigurd B. Angenent, Curve shortening and the topology of closed geodesics on surfaces, Ann. of Math. (2) 162 (2005), no. 3, 1187–1241. MR 2179729, DOI 10.4007/annals.2005.162.1187
Steven J. Altschuler, Singularities of the curve shrinking flow for space curves, J. Differential Geom. 34 (1991), no. 2, 491–514. MR 1131441
Paul Bryan and Janelle Louie, Classification of convex ancient solutions to curve shortening flow on the sphere, J. Geom. Anal. 26 (2016), no. 2, 858–872. MR 3472819, DOI 10.1007/s12220-015-9574-x
David L. Chopp, Computation of self-similar solutions for mean curvature flow, Experiment. Math. 3 (1994), no. 1, 1–15. MR 1302814, DOI 10.1080/10586458.1994.10504572
Tobias Holck Colding, William P. Minicozzi II, and Erik Kjær Pedersen, Mean curvature flow, Bull. Amer. Math. Soc. (N.S.) 52 (2015), no. 2, 297–333. MR 3312634, DOI 10.1090/S0273-0979-2015-01468-0
M. E. Gage, Curve shortening makes convex curves circular, Invent. Math. 76 (1984), no. 2, 357–364. MR 742856, DOI 10.1007/BF01388602
Michael E. Gage, Curve shortening on surfaces, Ann. Sci. École Norm. Sup. (4) 23 (1990), no. 2, 229–256. MR 1046497, DOI 10.24033/asens.1603
M. Gage and R. S. Hamilton, The heat equation shrinking convex plane curves, J. Differential Geom. 23 (1986), no. 1, 69–96. MR 840401, DOI 10.4310/jdg/1214439902
Yoshikazu Giga, Surface evolution equations, Monographs in Mathematics, vol. 99, Birkhäuser Verlag, Basel, 2006. A level set approach. MR 2238463
Hoeskuldur P. Halldorsson, Self-similar solutions to the curve shortening flow, Trans. Amer. Math. Soc. 364 (2012), no. 10, 5285–5309. MR 2931330, DOI 10.1090/S0002-9947-2012-05632-7
Hoeskuldur P. Halldorsson, Self-similar solutions to the mean curvature flow in the Minkowski plane $\Bbb {R}^{1,1}$, J. Reine Angew. Math. 704 (2015), 209–243. MR 3365779, DOI 10.1515/crelle-2013-0054
Tom Ilmanen and Brian White, Sharp lower bounds on density for area-minimizing cones, Camb. J. Math. 3 (2015), no. 1-2, 1–18. MR 3356355, DOI 10.4310/CJM.2015.v3.n1.a1
Naoyuki Ishimura, Curvature evolution of plane curves with prescribed opening angle, Bull. Austral. Math. Soc. 52 (1995), no. 2, 287–296. MR 1348488, DOI 10.1017/S0004972700014714
Carlo Mantegazza, Lecture notes on mean curvature flow, Progress in Mathematics, vol. 290, Birkhäuser/Springer Basel AG, Basel, 2011. MR 2815949, DOI 10.1007/978-3-0348-0145-4
W. W. Mullins, Two-dimensional motion of idealized grain boundaries, J. Appl. Phys. 27 (1956), 900–904. MR 78836, DOI 10.1063/1.1722511
John Urbas, Complete noncompact self-similar solutions of Gauss curvature flows. I. Positive powers, Math. Ann. 311 (1998), no. 2, 251–274. MR 1625754, DOI 10.1007/s002080050187