Mathematical Research Letters

Volume 26 (2019)

Number 5

Mean curvature flows of closed hypersurfaces in warped product manifolds

Pages: 1393 – 1413

DOI: https://dx.doi.org/10.4310/MRL.2019.v26.n5.a8

Authors

Zheng Huang (Department of Mathematics, City University of New York, Staten Island, N.Y., U.S.A.; and Graduate Center, City University of New York, N.Y., U.S.A.)

Zhou Zhang (School of Mathematics and Statistics, University of Sydney, NSW, Australia)

Hengyu Zhou (College of Mathematics and Statistics, Chongqing University, Chongqing, China)

Abstract

We investigate the mean curvature flows in a class of warped products manifolds with closed hypersurfaces fibering over $\mathbb{R}$. In particular, we prove that under some natural conditions on the warping function and Ricci curvature bound for the ambient space, there exists a large class of closed initial hypersurfaces, as geodesic graphs over the totally geodesic hypersurface $N$, such that the mean curvature flow starting from $S_0$ exists for all time and converges to $N$.

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Z.H. thanks a grant from the Simons Foundation (#359635), and the support from U.S. NSF grants DMS 1107452, 1107263, 1107367 “RNMS: Geometric Structures and Representation varieties” (the GEAR Network), Z.Z. is partially supported by Australian Research Council Future Fellowship FT150100341, and H.Z. is supported by the National Natural Science Foundation of China NSFC #11801046.

Received 15 April 2018

Accepted 10 February 2019

Published 27 November 2019