Mean curvature flow of graphs in warped products
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- by Alexander A. Borisenko and Vicente Miquel PDF
- Trans. Amer. Math. Soc. 364 (2012), 4551-4587 Request permission
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.References
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
- 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. - © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 364 (2012), 4551-4587
- MSC (2010): Primary 53C44; Secondary 53C40, 53C21
- DOI: https://doi.org/10.1090/S0002-9947-2012-05425-0
- MathSciNet review: 2922601
Dedicated: Dedicated to Professor Antonio M. Naveira on the occasion of his 70th birthday