Advertisement

The Journal of Geometric Analysis

, Volume 28, Issue 2, pp 1749–1772 | Cite as

Inverse Mean Curvature Flows in Warped Product Manifolds

  • Hengyu ZhouEmail author
Article

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.

Keywords

Inverse mean curvature flow Warped product manifold Asymptotic behavior 

Mathematics Subject Classification

Primary 53C44 Secondary 53C42 35J93 35B45 35K93 

Notes

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.

References

  1. 1.
    Allen, B.: Long time existence of non-compact inverse mean curvature flow in hyperbolic space (2015). arXiv:1510.06670
  2. 2.
    Besse, A.L.: Einstein Manifolds. Classics in Mathematics. Springer, Berlin (2008). (Reprint of the 1987 edition)Google Scholar
  3. 3.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Brendle, S.: Constant mean curvature surfaces in warped product manifolds. Publ. Math. Inst. Hautes Études Sci. 117, 247–269 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Castro, I., Lerma, A.M.: Homothetic solitons for the inverse mean curvature flow (2015). arXiv:1511.03826
  7. 7.
    Chen, L., Mao, J.: Nonaparmetric inverse curvature flows in the Ads-Schwarzschild manifold. J. Geom. Anal. (2017). doi: 10.1007/s12220-017-9848-6
  8. 8.
    Ding, Q.: The inverse mean curvature flow in rotationally symmetric spaces. Chin. Ann. Math. Ser. B 32(1), 27–44 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Ecker, K., Huisken, G.: Mean curvature evolution of entire graphs. Ann. of Math (2) 130(3), 453–471 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Gerhardt, C.: Flow of nonconvex hypersurfaces into spheres. J. Differ. Geom. 32(1), 299–314 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Gerhardt, C.: Inverse curvature flows in hyperbolic space. J. Differ. Geom. 89(3), 487–527 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Guan, P., Li, J.: A mean curvature type flow in space forms. Int. Math. Res. Not. 13, 4716–4740 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Guo, F., Li, G., Wu, C.: Isoperimetric inequalities for eigenvalues by inverse mean curvature flow (2016). arXiv:1602.05290
  14. 14.
    Hamilton, R.: Three-manifolds with positive Ricci curvature. J. Differ. Geom. 17, 255–306 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Hoffman, D., Spruck, J.: Sobolev and isoperimetric inequalities for Riemannian submanifolds. Commun. Pure Appl. Math. 27, 715–727 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Huang, Z., Lin, L., Zhang, Z.: Mean curvature flow in Fuchsian manifolds (2016). arXiv:1605.06565
  17. 17.
    Huisken, G.: Contracting convex hypersurfaces in Riemannian manifolds by their mean curvature. Invent. Math. 84(3), 463–480 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Huisken, G., Ilmanen, T.: The inverse mean curvature flow and the Riemannian Penrose inequality. J. Differ. Geom. 59(3), 353–437 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Huisken, G., Ilmanen, T.: Higher regularity of the inverse mean curvature flow. J. Differ. Geom. 80(2008), 433–452 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    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)Google Scholar
  22. 22.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    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
  24. 24.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Montiel, S.: Unicity of constant mean curvature hypersurfaces in some Riemannian manifolds. Indiana Univ. Math. J. 48(2), 711–748 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Mullins, T.: On the inverse mean curvature flow in warped product manifolds (2016). arXiv:1610.05234
  27. 27.
    Scheuer, J.: Pinching and asymptotical roundness for inverse curvature flows in Euclidean space. J. Geom. Anal. 26(3), 2265–2281 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Scheuer, J.: The inverse mean curvature flow in warped cylinders of non-positive radial curvature. Adv. Math. 306, 1130–1163 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Smoczyk, K.: Remarks on the inverse mean curvature flow. Asian J. Math. 29(2), 331–335 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Urbas, J.I.E.: On the expansion of starshaped hypersurfaces by symmetric functions of their principal curvatures. Math. Z. 205(205), 355–372 (1990)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 2017

Authors and Affiliations

  1. 1.Department of MathematicsSun Yat-sen UniversityGuangzhouPeople’s Republic of China

Personalised recommendations