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Soliton solutions of the mean curvature flow and minimal hypersurfaces


Authors: Norbert Hungerbühler and Thomas Mettler
Journal: Proc. Amer. Math. Soc. 140 (2012), 2117-2126
MSC (2010): Primary 49Q05
DOI: https://doi.org/10.1090/S0002-9939-2011-11205-X
Published electronically: October 21, 2011
MathSciNet review: 2888198
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $ (M,g)$ be an oriented Riemannian manifold of dimension at least $ 3$ and $ \mathbf {X} \in \mathfrak{X}(M)$ a vector field. We show that the Monge-Ampère differential system (M.A.S.) for $ \mathbf {X}$-pseudosoliton hypersurfaces on $ (M,g)$ is equivalent to the minimal hypersurface M.A.S. on $ (M,\bar {g})$ for some Riemannian metric $ \bar {g}$ if and only if $ \mathbf {X}$ is the gradient of a function $ u$, in which case $ \bar {g}=e^{-2u}g$. Counterexamples to this equivalence for surfaces are also given.


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  • 1. Juan-Carlos Álvarez-Paiva and Gautier Berck, Finsler surfaces with prescribed geodesics, arXiv:math/1002.0243v1, 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

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Additional Information

Norbert Hungerbühler
Affiliation: Department of Mathematics, ETH Zürich, CH-8092 Zurich, Switzerland
Email: norbert.hungerbuehler@math.ethz.ch

Thomas Mettler
Affiliation: Department of Mathematics, University of California at Berkeley, Berkeley, California 94720
Email: mettler@math.berkeley.edu

DOI: https://doi.org/10.1090/S0002-9939-2011-11205-X
Keywords: Mean curvature flow, soliton solutions, minimal hypersurfaces, Monge-Ampère systems, equivalence problem
Received by editor(s): February 11, 2011
Published electronically: October 21, 2011
Additional Notes: Research for this article was carried out while the authors were supported by the Swiss National Science Foundation, the first author by the grant 200020-124668, and the second by the postdoctoral fellowship PBFRP2-133545.
Communicated by: Jianguo Cao
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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