Skip to main content
SpringerLink
Log in

An Extension of Hsiung–Minkowski Formulas and Some Applications

  • Published:
The Journal of Geometric Analysis Aims and scope Submit manuscript

Abstract

We prove a generalization of Hsiung–Minkowski formulas for closed submanifolds in semi-Riemannian manifolds with constant curvature. As a corollary, we obtain volume and area upper bounds for \(k\)-convex hypersurfaces in terms of a weighted total \(k\)-th mean curvature of the hypersurface. We also obtain some Alexandrov-type results and some eigenvalue estimates for hypersurfaces.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aledo, J.A., Alías, L.J., Romero, A.: Integral formulas for compact space-like hypersurfaces in de Sitter space: applications to the case of constant higher order mean curvature. J. Geom. Phys. 31(2), 195–208 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  2. Alencar, H., Colares, A.G.: Integral formulas for the \(r\)-mean curvature linearized operator of a hypersurface. Ann. Glob. Anal. Geom. 16(3), 203–220 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  3. Alencar, H., do Carmo, M., Rosenberg, H.: On the first eigenvalue of the linearized operator of the \(r\)-th mean curvature of a hypersurface. Ann. Glob. Anal. Geom. 11(4), 387–395 (1993)

    Article  MATH  Google Scholar 

  4. Alías, L.J., Brasil Jr. A., Colares, A.G.: Integral formulae for spacelike hypersurfaces in conformally stationary spacetimes and applications. Proc. Edinb. Math. Soc. 46(2), 465–488 (2003)

  5. Barbosa, J.L., Colares, A.G.: Stability of hypersurfaces with constant-mean curvature. Ann. Glob. Anal. Geom. 15(3), 277–297 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  6. Brendle, S.: Constant mean curvature surfaces in warped product manifolds. Publ. Math. de l’IHÉS 117(1), 247–269 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  7. Chen, B.-Y., Yano, K.: Integral formulas for submanifolds and their applications. J. Differ. Geom. 5(3–4), 467–477 (1971)

    MATH  MathSciNet  Google Scholar 

  8. Cheng, Q.-M., Ishikawa, S.: Spacelike hypersurfaces with constant scalar curvature. Manuscr. Math. 95(4), 499–505 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  9. De Lellis, C., Topping, P.M.: Almost-Schur lemma. Calc. Var. Partial Differ. Equ. 43, 347–354 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  10. Garay, O.J.: An application of Reilly’s formula. Bull. Lond. Math. Soc. 21(2), 176–178 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  11. Garding, L.: An inequality for hyperbolic polynomials. J. Math. Mech. 8(6), 957–965 (1959)

    MATH  MathSciNet  Google Scholar 

  12. Grosjean, J.F.: Upper bounds for the first eigenvalue of the Laplacian on compact submanifolds. Pac. J. Math. 206(1), 93–112 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  13. Heintze, E.: Extrinsic upper bounds for \(\lambda _1\). Math. Ann. 280(3), 389–402 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  14. Hsiung, C.-C.: Some integral formulas for closed hypersurfaces in Riemannian space. Pac. J. Math. 6(2), 291–299 (1956)

    Article  MATH  MathSciNet  Google Scholar 

  15. Koh, S.-E.: A characterization of round spheres. Proc. Am. Math. Soc. 126(12), 3657–3660 (1998)

    Article  MATH  Google Scholar 

  16. Koh, S.-E.: Sphere theorem by means of the ratio of mean curvature functions. Glasg. Math. J. 42(1), 91–95 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  17. Koh, S.-E., Lee, S.-W.: Addendum to the paper: sphere theorem by means of the ratio of mean curvature functions. Glasg. Math. J. 43(2), 275–276 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  18. Koiso, N.: Hypersurfaces of Einstein manifolds. In: Annales scientifiques de l’École Normale Supérieure, vol. 14, pp. 433–443. Société mathématique de France, Paris (1981)

  19. Kwong, K.-K.: On an inequality of Andrews, De Lellis, and Topping. J. Geom. Anal. 1–14 (2012)

  20. Kwong, K.-K.: On convex hypersurfaces in space forms and eigenvalues estimates for differential forms. arXiv:1207.3999 (2012)

  21. 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)

    Article  MATH  MathSciNet  Google Scholar 

  22. Montiel, S., Ros, A.: Compact hypersurfaces: the Alexandrov theorem for higher order mean curvatures. Differ. Geom. 52, 279–296 (1991)

    MathSciNet  Google Scholar 

  23. Raulot, S., Savo, A.: On the first eigenvalue of the Dirichlet-to-Neumann operator on forms. J. Funct. Anal. 262(3), 889–914 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  24. Reilly, R.C.: Variational properties of functions of the mean curvatures for hypersurfaces in space forms. J. Differ. Geom. 8, 465–477 (1973)

    MATH  MathSciNet  Google Scholar 

  25. Ros, A.: Compact hypersurfaces with constant higher order mean curvatures. Rev. Mat. Iberoam. 3(3), 447–453 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  26. Shetty, D.J., Amur, K.: Integral formulas for submanifolds and their applications. J. Differ. Geom. 15(4), 513–529 (1980)

    MATH  MathSciNet  Google Scholar 

  27. Strübing, W.: On integral formulas for submanifolds of spaces of constant curvature and some applications. Manuscr. Math. 49(2), 177–194 (1984)

    Article  MATH  Google Scholar 

  28. Veeravalli, A.R.: On the first Laplacian eigenvalue and the center of gravity of compact hypersurfaces. Comment. Math. Helv. 76(1), 155–160 (2001)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

This work was conducted when the author was working as a Research Fellow at Monash University. He would like to thank Monash University for providing an excellent research environment.

Author information

Authors and Affiliations

  1. Department of Mathematics, National Cheng Kung University, Tainan, Taiwan, ROC

    Kwok-Kun Kwong

Authors
  1. Kwok-Kun Kwong
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kwok-Kun Kwong.

Additional information

Communicated by Ben Andrews.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kwong, KK. An Extension of Hsiung–Minkowski Formulas and Some Applications. J Geom Anal 26, 1–23 (2016). https://doi.org/10.1007/s12220-014-9536-8

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12220-014-9536-8

Keywords

Mathematics Subject Classification

Search

Navigation