Inverse Mean Curvature Flows in Warped Product Manifolds
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We study inverse mean curvature flows of starshaped, mean convex hypersurfaces in warped product manifolds with a positive warping factor \(\phi (r)\). If \(\phi '(r)>0\) and \(\phi ''(r)\ge 0\), we show that these flows exist for all times, remain starshaped, and mean convex. Plus the positivity of \(\phi ''(r)\) and a curvature condition we obtain a lower positive bound of mean curvature along these flows independent of the initial mean curvature. We also give a sufficient condition to extend the asymptotic behavior of these flows in Euclidean spaces into some more general warped product manifolds.
KeywordsInverse mean curvature flow Warped product manifold Asymptotic behavior
Mathematics Subject ClassificationPrimary 53C44 Secondary 53C42 35J93 35B45 35K93
This work was supported by the National Natural Science Foundation of China, Nos. 11261378 and 11521101. A portion of this work was done when the author visited Nanjing University from September 2015 to March 2016. The author has greatly profited for discussions with Prof. Jiaqiang Mei, Prof. Yalong Shi and Prof. Yiyan Xu. He is very grateful to encouragements from Prof. Zheng Huang, Prof. Yunping Jiang and Prof. Lixin Liu. It is a pleasure to thank the comments from Prof. Yong Wei and Prof. Mat Langford. The author also thanks the referees for careful readings and helpful suggestions.
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