Soliton Solutions of the Mean Curvature Flow and Minimal Hypersurfaces

References

[1] Juan Carlos Alvarez-Paiva and Gautier Berck, Finsler surfaces with prescribed geodesics, arXiv:math/1002.0242v1, 2010.

[2] Sigurd Angenent, On the formation of singularities in the curve shortening flow, J. Differential Geom. 33 (1991), no. 3, 601–633. MR 1100205

[3] Sigurd B. Angenent, Shrinking doughnuts, Nonlinear diffusion equations and their equilibrium states, 3 (Gregynog, 1989), Progr. Nonlinear Differential Equations Appl., vol. 7, Birkhäuser Boston, Boston, MA, 1992, pp. 21–38. MR 1167827

[4] Robert Bryant, Maciej Dunajski, and Michael Eastwood, Metrisability of two-dimensional projective structures, J. Differential Geom. 83 (2009), no. 3, 465–499. MR 2581355

[5] Robert Bryant, Phillip Griffiths, and Daniel Grossman, Exterior differential systems and Euler-Lagrange partial differential equations, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 2003. MR 1985469

[6] Julie Clutterbuck, Oliver C. Schnürer, and Felix Schulze, Stability of translating solutions to mean curvature flow, Calc. Var. Partial Differential Equations 29 (2007), no. 3, 281–293. MR 2321890

[7] Gerhard Huisken, Asymptotic behavior for singularities of the mean curvature flow, J. Differential Geom. 31 (1990), no. 1, 285–299. MR 1030675

[8] N. Hungerbühler and K. Smoczyk, Soliton solutions for the mean curvature flow, Differential Integral Equations 13 (2000), no. 10-12, 1321–1345. MR 1787070

[9] Norbert Hungerbühler and Beatrice Roost, Mean curvature flow solitons, Analytic aspects of problems in Riemannian geometry: Elliptic PDEs, solitons and computer imaging, Séminaires et Congrès, vol. 19, Société Mathématiqe de France, 2009, pp. 129–158.

[10] Paulette Libermann and Charles-Michel Marle, Symplectic geometry and analytical mechanics, Mathematics and its Applications, vol. 35, D. Reidel Publishing Co., Dordrecht, 1987, Translated from the French by Bertram Eugene Schwarzbach. MR 882548

[11] V. V. Lychagin, V. N. Rubtsov, and I. V. Chekalov, A classification of Monge-Ampère equations, Ann. Sci. École Norm. Sup. (4) 26 (1993), no. 3, 281–308. MR 1222276

[12] Thomas Mettler, On the Weyl metrisability problem for projective surfaces and related topics, Ph.D. thesis, Université de Fribourg, 2010.

[13] Knut Smoczyk, A relation between mean curvature flow solitons and minimal submanifolds, Math. Nachr. 229 (2001), 175–186. MR 1855161

  1. Since the computations are somewhat complex, they have been carried out using maple. The maple file can be obtained from the authors upon request.

  2. Stated on the occasion of a seminar talk of the first author at the Institute for Mathematical Research (FIM) at ETH Zürich, March 1999.

  3. More generally one can define a M.A.S. to be a differential ideal which is only locally generated by a contact ideal and an \(n\)-form. However for our purposes the above definition is sufficient.