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

MathSciNet review:
2922601

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a complete Riemannian manifold which is either compact or has a pole, and let be a positive smooth function on . In the warped product , we study the flow by the mean curvature of a locally Lipschitz continuous graph on and prove that the flow exists for all time and that the evolving hypersurface is for and is a graph for all . 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

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.