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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Mean curvature flow of graphs in warped products


Authors: Alexander A. Borisenko and Vicente Miquel
Journal: Trans. Amer. Math. Soc. 364 (2012), 4551-4587
MSC (2010): Primary 53C44; Secondary 53C40, 53C21
Published electronically: April 11, 2012
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Abstract: Let $ M$ be a complete Riemannian manifold which is either compact or has a pole, and let $ \varphi $ be a positive smooth function on $ M$. In the warped product $ M\times _\varphi \mathbb{R}$, we study the flow by the mean curvature of a locally Lipschitz continuous graph on $ M$ and prove that the flow exists for all time and that the evolving hypersurface is $ C^\infty $ for $ t>0$ and is a graph for all $ t$. Moreover, under certain conditions, the flow has a well-defined limit.


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Additional Information

Alexander A. Borisenko
Affiliation: Mathematics Faculty, Geometry Department, Kharkov National University, Pl. Svobodi 4, 61077-Kharkov, Ukraine
Email: borisenk@univer.kharkov.ua

Vicente Miquel
Affiliation: Departamento de Geometría y Topología, Universidad de Valencia, Avda. Andrés Estellés 1, 46100-Burjassot (Valencia) Spain
Email: miquel@uv.es

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05425-0
PII: S 0002-9947(2012)05425-0
Keywords: Differential geometry, algebraic geometry
Received by editor(s): January 30, 2009
Received by editor(s) in revised form: July 12, 2010
Published electronically: April 11, 2012
Additional Notes: This work was done while the first author was Visiting Professor at the University of Valencia in 2008, supported by a \lq\lq ayuda del Ministerio de Educación y Ciencia SAB2006-0073.” He wants to thank that university and its Department of Geometry and Topology for the facilities they gave him.
The second author was partially supported by the DGI(Spain) and FEDER Project MTM2010-1544 and the Generalitat Valenciana Project Prometeo 2009/099
Both authors want to thank the referee for pointing out a mistake in a previous version of the paper.
Dedicated: Dedicated to Professor Antonio M. Naveira on the occasion of his 70th birthday
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.