Skip to main content
SpringerLink
Log in

A Bernstein Theorem for the Abreu Equation

  • Published:
Results in Mathematics Aims and scope Submit manuscript

Abstract

In this paper we prove a Bernstein type theorem for the Abreu equation on a complete Riemannian manifold (M,G *). Using this theorem and affine blow-up analysis we obtain interior estimates for the Abreu equation.

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. Chen, B.H., Li, A.-M., Sheng, L.: Affine techniques on extremal metrics on toric surface. Preprint. ArXiv:1008.2607

  2. Delzant T.: Hamiltioniens periodiques et imageconvex de l’application moment. Bull. Soc. Math. France 116, 315–339 (1988)

    MathSciNet  MATH  Google Scholar 

  3. Donaldson S.K.: Extremal metrics on toric surfaces:a continuity method. J. Diff. Geom. 79, 389–432 (2008)

    MathSciNet  MATH  Google Scholar 

  4. Gutiérrez C.E.: The Monge–Ampère equations. In: Progress in Nonlinear Differential Equations and Applications, Vol. 44. Birkhauser, Basel (2001)

    Google Scholar 

  5. Li, A.-M.: The Estimates for Abreu Equation. Sichuan University, Sichuan (2009). Preprint

  6. Li A.-M., Xu R.W.: A rigidity theorem for Affine Kähler–Ricci flat graph. Result. Math. 56, 141–164 (2009)

    Article  MATH  Google Scholar 

  7. Li A.-M., Jia F.: A Bernstein property ofsome fourth order partial differential equations. Result. Math. 56, 109–139 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  8. Li A.-M., Jia F.: A Bernstein property of affine maximal hypersurfaces. Ann. Glob. Anal. Geom. 23, 359–372 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  9. Li A.-M., Jia F.: Euclidean complete affine surfaces with constantaffine mean curvature. Ann. Glob. Anal. Geom. 23, 283–304 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  10. Li A.-M., Jia F.: The Calabi conjectureon affine maximal surfaces. Result. Math. 40, 265–272 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  11. Li A.-M., Xu R.W., Simon U., Jia F.: Bernstein Problems and Monge–Ampère Equations. World Scientific, Hackensack (2010)

    Book  MATH  Google Scholar 

  12. Li A.-M., Xu R.W., Simon U., Jia F.: on Chern’s affine Bernstein conjecture. Result. Math. 60, 133–155 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  13. Jia, F., Li, A.-M.: Complete affine Kähler manifolds. Preprint. ArXiv:1008.2604

  14. Sheng L., Chen G.: A Bernstein property of some affine Kähler scalar flat graph. Result. Math. 56, 165–175 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Trudinger N., Wang X.: The Bernstein problem for affine maximal hypersurfaces. Invent. Math. 140, 399–422 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  16. Xu R.W.: Bernstein properties for some relative parabolicaffine hyperspheres. Result. Math. 52, 409–422 (2008)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of Mathematics, Sichuan University, Chengdu, 610064, Sichuan, People’s Republic of China

    Min Xiong & Li Sheng

  2. College of Science, Southwest University of Science and Technology, Mianyang, 621000, Sichuan, People‘s Republic of China

    Min Xiong

Authors
  1. Min Xiong
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Li Sheng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Li Sheng.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Xiong, M., Sheng, L. A Bernstein Theorem for the Abreu Equation. Results. Math. 63, 1195–1207 (2013). https://doi.org/10.1007/s00025-012-0262-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00025-012-0262-x

Mathematics Subject Classification (2000)

Keywords

Search

Navigation