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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
Published electronically: October 21, 2011
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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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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: http://dx.doi.org/10.1090/S0002-9939-2011-11205-X
PII: S 0002-9939(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.