We study the area preserving curve shortening flow with Neumann free boundary conditions outside of a convex domain in the Euclidean plane. Under certain conditions on the initial curve the flow does not develop any singularity, and it subconverges smoothly to an arc of a circle sitting outside of the given fixed domain and enclosing the same area as the initial curve.
@article{bwmeta1.element.doi-10_1515_geofl-2015-0004,
author = {Elena M\"ader-Baumdicker},
title = {The area preserving curve shortening flow with Neumann free boundary conditions},
journal = {Geometric Flows},
volume = {1},
year = {2015},
zbl = {1297.53004},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_1515_geofl-2015-0004}
}
Elena Mäder-Baumdicker. The area preserving curve shortening flow with Neumann free boundary conditions. Geometric Flows, Tome 1 (2015) . http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_1515_geofl-2015-0004/
[1] U. Abresch and J. Langer. The normalized curve shortening flow and homothetic solutions. J. Differential Geom., 23(2):175– 196, 1986. | Zbl 0592.53002
[2] M. Athanassenas. Volume-preserving mean curvature flow of rotationally symmetric surfaces. Comment. Math. Helv., 72(1):52–66, 1997. [Crossref] | Zbl 0873.35033
[3] M. Athanassenas. Behaviour of singularities of the rotationally symmetric, volume-preserving mean curvature flow. Calc. Var. Partial Differential Equations, 17(1):1–16, 2003. | Zbl 1045.53045
[4] M. Athanassenas and S. Kandanaarachchi. Convergence of axially symmetric volume-preserving mean curvature flow. Pacific J. Math., 259(1):41–54, 2012. | Zbl 1258.53067
[5] T. Aubin. Some nonlinear problems in Riemannian geometry. Springer Monographs inMathematics. Springer-Verlag, Berlin, 1998.
[6] K. A. Brakke. The motion of a surface by its mean curvature, volume 20 of Mathematical Notes. Princeton University Press, Princeton, N.J., 1978. | Zbl 0386.53047
[7] J. A. Buckland. Mean curvature flow with free boundary on smooth hypersurfaces. J. Reine Angew. Math., 586:71–90, 2005. | Zbl 1082.37043
[8] E. Cabezas-Rivas and V. Miquel. Volume-preserving mean curvature flow of revolution hypersurfaces in a rotationally symmetric space. Math. Z., 261(3):489–510, 2009. [WoS] | Zbl 1161.53053
[9] E. Cabezas-Rivas and V. Miquel. Volume preserving mean curvature flow of revolution hypersurfaces between two equidistants. Calc. Var. Partial Differential Equations, 43(1-2):185–210, 2012. | Zbl 1247.53080
[10] J. Choe, M. Ghomi, and M. Ritoré. The relative isoperimetric inequality outside convex domains in Rn. Calc. Var. Partial Differential Equations, 29(4):421–429, 2007. | Zbl 1116.58016
[11] M. P. do Carmo. Differential geometry of curves and surfaces. Prentice-Hall Inc., Englewood Cliffs, N.J., 1976. | Zbl 0326.53001
[12] G. Dziuk, E. Kuwert, and R. Schätzle. Evolution of elastic curves in Rn: existence and computation. SIAM J. Math. Anal., 33(5):1228–1245 (electronic), 2002. | Zbl 1031.53092
[13] K. Ecker. Regularity theory for mean curvature flow. Progress in Nonlinear Differential Equations and their Applications, 57. Birkhäuser Boston Inc., Boston, MA, 2004. | Zbl 1058.53054
[14] A. Friedman. Partial differential equations of parabolic type. Prentice-Hall Inc., Englewood Cliffs, N.J., 1964. | Zbl 0144.34903
[15] M. Gage. On an area-preserving evolution equation for plane curves. In Nonlinear problems in geometry (Mobile, Ala., 1985), volume 51 of Contemp. Math., pages 51–62. Amer. Math. Soc., Providence, RI, 1986.
[16] H. P. Halldorsson. Self-similar solutions to the curve shortening flow. Trans. Amer. Math. Soc., 364(10):5285–5309, 2012. | Zbl 1281.53066
[17] R. S. Hamilton. Four-manifolds with positive curvature operator. J. Differential Geom., 24(2):153–179, 1986. | Zbl 0628.53042
[18] R. S. Hamilton. The formation of singularities in the Ricci flow. In Surveys in differential geometry, Vol. II (Cambridge, MA, 1993), pages 7–136. Int. Press, Cambridge, MA, 1995. | Zbl 0867.53030
[19] R. S. Hamilton. Harnack estimate for the mean curvature flow. J. Differential Geom., 41(1):215–226, 1995. | Zbl 0827.53006
[20] G. Huisken. The volume preserving mean curvature flow. J. Reine Angew. Math., 382:35–48, 1987. | Zbl 0621.53007
[21] G. Huisken. Asymptotic behavior for singularities of the mean curvature flow. J. Differential Geom., 31(1):285–299, 1990. | Zbl 0694.53005
[22] G. Huisken and C. Sinestrari. Mean curvature flow singularities for mean convex surfaces. Calc. Var. Partial Differential Equations, 8(1):1–14, 1999. | Zbl 0992.53052
[23] W. Klingenberg. Eine Vorlesung über Differentialgeometrie. Springer-Verlag, Berlin, 1973. Heidelberger Taschenbücher, Band 107. | Zbl 0267.53001
[24] A. Ludwig. A Relaxed Partitioning Disk for Strictly Convex Domains. PhD thesis, Freiburg: Univ. Freiburg, Fac. of Math. and Phys. 94 S., 2013. | Zbl 1278.49001
[25] E. Mäder-Baumdicker. The area preserving curve shortening flow with Neumann free boundary conditions. PhD thesis, Freiburg: Univ. Freiburg, Fac. of Math. and Phys. 155 S., 2014. | Zbl 1297.53004
[26] A. Magni and C. Mantegazza. A Note on Grayson’s Theorem. Rend. Sem. Mat. Univ. Padova, 131:263–279, 2014. | Zbl 1296.53133
[27] C. Mantegazza, M. Novaga, and V. M. Tortorelli. Motion by curvature of planar networks. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 3(2):235–324, 2004. | Zbl 1170.53313
[28] M. H. Protter and H. F. Weinberger. Maximum principles in differential equations. Prentice-Hall Inc., Englewood Cliffs, N.J., 1967. | Zbl 0153.13602
[29] M. Ritoré and C. Sinestrari. Mean curvature flow and isoperimetric inequalities. Advanced Courses in Mathematics. CRM Barcelona. Birkhäuser Verlag, Basel, 2010. Edited by Vicente Miquel and Joan Porti. | Zbl 1189.53002
[30] A. Stahl. On the mean curvature flow with Neumann boundary values on smooth hypersurfaces. (Über den mittleren Krümmungsfluß mit Neumannrandwerten auf glatten Hyperflächen.). PhD thesis, Tübingen: Univ. Tübingen, Math. Fak. 129 S., 1994.
[31] A. Stahl. Convergence of solutions to the mean curvature flow with a Neumann boundary condition. Calc. Var. Partial Differential Equations, 4(5):421–441, 1996. | Zbl 0896.35059
[32] A. Stahl. Regularity estimates for solutions to the mean curvature flow with a Neumann boundary condition. Calc. Var. Partial Differential Equations, 4(4):385–407, 1996. | Zbl 0851.35053
[33] A. Stone. A density function and the structure of singularities of the mean curvature flow. Calc. Var. Partial Differential Equations, 2(4):443–480, 1994. | Zbl 0833.35062
[34] A.Windel. Über den volumenerhaltenden mittleren Krümmungsfluß mit einer Neumann-Randbedingung auf einer konvexen Hyperfläche. Diplom thesis, Univ. Freiburg, Fac. of Math. and Phys., 100 S., 2006.