Abstract
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.
Similar content being viewed by others
References
Allen, B.: Long time existence of non-compact inverse mean curvature flow in hyperbolic space (2015). arXiv:1510.06670
Besse, A.L.: Einstein Manifolds. Classics in Mathematics. Springer, Berlin (2008). (Reprint of the 1987 edition)
Bray, H.L., Neves, A.: Classification of prime 3-manifolds with Yamabe invariant greater than \(\mathbb{R} \mathbb{P}^3\). Ann. Math. 159(1), 407–424 (2004)
Brendle, S.: Constant mean curvature surfaces in warped product manifolds. Publ. Math. Inst. Hautes Études Sci. 117, 247–269 (2013)
Brendle, S., Hung, P.-K., Wang, M.-T.: A Minkowski inequality for hypersurfaces in the anti-de Sitter-Schwarzschild manifold. Commun. Pure Appl. Math. 69(1), 124–144 (2016)
Castro, I., Lerma, A.M.: Homothetic solitons for the inverse mean curvature flow (2015). arXiv:1511.03826
Chen, L., Mao, J.: Nonaparmetric inverse curvature flows in the Ads-Schwarzschild manifold. J. Geom. Anal. (2017). doi:10.1007/s12220-017-9848-6
Ding, Q.: The inverse mean curvature flow in rotationally symmetric spaces. Chin. Ann. Math. Ser. B 32(1), 27–44 (2011)
Ecker, K., Huisken, G.: Mean curvature evolution of entire graphs. Ann. of Math (2) 130(3), 453–471 (1989)
Gerhardt, C.: Flow of nonconvex hypersurfaces into spheres. J. Differ. Geom. 32(1), 299–314 (1990)
Gerhardt, C.: Inverse curvature flows in hyperbolic space. J. Differ. Geom. 89(3), 487–527 (2011)
Guan, P., Li, J.: A mean curvature type flow in space forms. Int. Math. Res. Not. 13, 4716–4740 (2015)
Guo, F., Li, G., Wu, C.: Isoperimetric inequalities for eigenvalues by inverse mean curvature flow (2016). arXiv:1602.05290
Hamilton, R.: Three-manifolds with positive Ricci curvature. J. Differ. Geom. 17, 255–306 (1982)
Hoffman, D., Spruck, J.: Sobolev and isoperimetric inequalities for Riemannian submanifolds. Commun. Pure Appl. Math. 27, 715–727 (1974)
Huang, Z., Lin, L., Zhang, Z.: Mean curvature flow in Fuchsian manifolds (2016). arXiv:1605.06565
Huisken, G.: Contracting convex hypersurfaces in Riemannian manifolds by their mean curvature. Invent. Math. 84(3), 463–480 (1986)
Huisken, G., Ilmanen, T.: The inverse mean curvature flow and the Riemannian Penrose inequality. J. Differ. Geom. 59(3), 353–437 (2001)
Huisken, G., Ilmanen, T.: Higher regularity of the inverse mean curvature flow. J. Differ. Geom. 80(2008), 433–452 (2008)
Hung, P.-K., Wang, M.-T.: Inverse mean curvature flows in the hyperbolic 3-space revisited. Calc. Var. Partial Differ. Equ. 54(1), 119–126 (2015)
Krylov, N.V.: Nonlinear elliptic and parabolic equations of the second order. Sold and distributed in the U.S.A. and Canada. Kluwer Academic Publishers, Norwell (1987)
Kwong, K.-K., Miao, P.: A new monotone quantity along the inverse mean curvature flow in \(\mathbb{R}^n\). Pac. J. Math. 267(2), 417–422 (2014)
Li, H., Wei, Y.: On the inverse mean curvature flow in Schwarzschild space and Kottler space. Calc. Var. 56, 62 (2017). doi:10.1007/s00526-017-1160-6
Michael, J.H., Simon, L.M.: Sobolev and mean-value inequalities on generalized submanifolds of \(R^{n}\). Commun. Pure Appl. Math. 26, 361–379 (1973)
Montiel, S.: Unicity of constant mean curvature hypersurfaces in some Riemannian manifolds. Indiana Univ. Math. J. 48(2), 711–748 (1999)
Mullins, T.: On the inverse mean curvature flow in warped product manifolds (2016). arXiv:1610.05234
Scheuer, J.: Pinching and asymptotical roundness for inverse curvature flows in Euclidean space. J. Geom. Anal. 26(3), 2265–2281 (2016)
Scheuer, J.: The inverse mean curvature flow in warped cylinders of non-positive radial curvature. Adv. Math. 306, 1130–1163 (2017)
Smoczyk, K.: Remarks on the inverse mean curvature flow. Asian J. Math. 29(2), 331–335 (2000)
Urbas, J.I.E.: On the expansion of starshaped hypersurfaces by symmetric functions of their principal curvatures. Math. Z. 205(205), 355–372 (1990)
Acknowledgements
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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Zhou, H. Inverse Mean Curvature Flows in Warped Product Manifolds. J Geom Anal 28, 1749–1772 (2018). https://doi.org/10.1007/s12220-017-9887-z
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12220-017-9887-z