Skip to main content
Log in

The Gauss–Bonnet–Chern mass for graphic manifolds

  • Published:
Annals of Global Analysis and Geometry Aims and scope Submit manuscript

Abstract

In this paper, we prove a positive mass theorem and Penrose-type inequality of the Gauss–Bonnet–Chern mass \(m_2\) for the graphic manifold with flat normal bundle.

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

Access this article

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

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Arnowitt, R., Deser, S., Misner, C.W.: Coordinate invariance and energy expressions in general relativity. Phys. Rev. (2) 122, 997–1006 (1961)

    Google Scholar 

  2. Bartnik, R.: The mass of an asymptotically flat manifold. Comm. Pure Appl. Math. 34, 661–693 (1986)

    Article  MathSciNet  Google Scholar 

  3. Bray, H.L., Lee, D.A.: On the Riemannian Penrose inequality in dimensions less than eight. Duke Math. J. 148(1), 81–106 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bray, H.L.: Proof of the Riemannian Penrose inequality using the positive mass theorem. J. Differ. Geom. 59(2), 177–267 (2001)

    MATH  MathSciNet  Google Scholar 

  5. Bray, H.L.: On the positive mass, Penrose, an ZAS inequalities in general dimension. Surveys in Geometric Analysis and Relativity. Adv. Lect. Math. (ALM), vol 20. Int. Press, Somerville (2011)

  6. Brendle, S., Hung, P.-K., Wang, M.-T.: A Minkowski-type inequality for hypersurfaces in the Anti-deSitter-Schwarzschild manifold, arXiv:1209.0669

  7. Chang, S.-Y.A., Wang, Y.: On Aleksandrov–Fenchel inequalities for k-convex domains. Milan J. Math. 79(1), 13–38 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  8. Davis, S.: Generalized Israel junction conditions for a Gauss–Bonnet brane world. Phys. Rev. D 67, 024030 (2003)

    Article  MathSciNet  Google Scholar 

  9. de Lima, L., Girão, F.: The ADM mass of asymptotically flat hypersurfaces. Trans. A.M.S. arXiv:1108.5474 (to appear)

  10. de Lima, L., Girão, F.: A rigidity result for the graph case of the Penrose inequality arXiv:1205.1132

  11. de Lima, L., Girão, F.: An Alexandrov-Fenchel-type ineuqality in hyperbolic space with an application to a Penrose inequality, arXiv:1209.0438

  12. Deser, S., Tekin, B.: Gravitational energy in quadratic-curvature gravities. Phys. Rev. Lett. 89, 101101 (2002)

    Article  MathSciNet  Google Scholar 

  13. Deser, S., Tekin, B.: Energy in generic higher curvature gravity theories. Phys. Rev. D 75, 084032 (2003)

    Article  MathSciNet  Google Scholar 

  14. Ge, Y., Wang, G., Wu, J.: A new mass for asymptotically flat manifolds, arXiv:1211.3645

  15. Ge, Y., Wang, G., Wu, J.: A positive mass theorem in the Einstein-Gauss-Bonnet theory, arXiv:1211.7305

  16. Ge, Y., Wang, G., Wu, J.: The Gauss–Bonnet–Chern mass of conformally flat manifolds. Int. Math. Res. Notices. arXiv:1212.3213 (to appear)

  17. Ge, Y., Wang, G., Wu, J.: Hyperbolic Alexandrov–Fenchel inequalities II. arXiv:1303.1714.1417

  18. Guan, P., Li, J.: The quermassintegral inequalities for k-convex starshaped domains. Adv. Math. 221, 1725–1732 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  19. Huang, L.-H., Wu, D.: Hypersurfaces with nonnegative scalar curvature. J. Differ. Geom. 95, 249–278 (2013)

    Google Scholar 

  20. Huang, L.-H., Wu, D.: The equality case of the Penrose inequality for asymptotically flat graphs. Trans. Amer. Math. Soc. arXiv:1205.2061 (to appear)

  21. Huisken, G., Ilmanen, T.: The inverse mean curvature flow and the Riemannian Penrose inequality. J. Differ. Geom. 59(3), 353–437 (2001)

    MATH  MathSciNet  Google Scholar 

  22. Lam, M.-K. G.: The graph cases of the Riemannian positive mass and Penrose inequality in all dimensions. arXiv.org/1010.4256

  23. Li, H., Wei, Y., Xiong, C.: A geometric inequality on hypersurface in hyperbolic space. arXiv:1211.4109

  24. Mars, M.: Topical review: present status of the Penrose inequality. Class. Quantum Gravity 26(19), 193001 (2009)

    Article  MathSciNet  Google Scholar 

  25. Mirandola, H., Vitorio, F.: The positive mass Theorem and Penrose inequality for graphical manifolds. arXiv:1304.3504

  26. Schoen, R., Yau, S.-T.: On the proof of the positive mass conjecture in general relativity. Comm. Math. Phys. 65(1), 45–76 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  27. Schoen, R., Yau, S.-T.: Proof of the Positive Mass Theorem II. Comm. Math. Phys. 79, 231–260 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  28. Schoen, R., Yau, S.T.: Conformally flat manifolds, Kleinian groups and scalar curvature. Invent. math. 92, 47–71 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  29. Wang, G., Xia, C.: Isoperimetric type problems and Alexandrov-Fenchel type inequalities in the hyperbolic space, arXiv:1304.1674

  30. Witten, E.: A new proof of the positive energy theorem. Comm. Math. Phys. 80, 381–402 (1981)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Yong Wei.

Additional information

The research of the authors was supported by NSFC No. 11271214.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Li, H., Wei, Y. & Xiong, C. The Gauss–Bonnet–Chern mass for graphic manifolds. Ann Glob Anal Geom 45, 251–266 (2014). https://doi.org/10.1007/s10455-013-9399-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10455-013-9399-4

Keywords

Mathematics Subject Classification (1991)

Navigation