Skip to main content
SpringerLink
Log in

A class of fully nonlinear equations arising from conformal geometry

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

In this paper we study the long time existence of solutions for a class of fully nonlinear parabolic equations arising from conformal geometry. In particular we prove that every smooth compact n dimensional manifold, \(n\ge3\), admits a Riemannian metric g with its Ricci curvature Ric and scalar curvature R satisfying

$$det(Ric-R\cdot g)=const..$$

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. Brendle S., Viaclovsky J. (2004) A variational characterization for σ n/2. Calc. Var. PDE. 20, 399–402

    MATH  MathSciNet  Google Scholar 

  2. Brooks R. (1989) A construction of metrics of negative Ricci curvature. J. Differential Geom. 29, 85–94

    MATH  MathSciNet  Google Scholar 

  3. Chang, A., Gursky, M., Yang, P. An equation of Monge-Ampère type in conformal geometry, and four manifolds of positive Ricci curvature. Ann. of Math.(2), 155, 709–789 (2002)

    Google Scholar 

  4. Chang, A., Gursky, M., Yang, P. An a priori estimate for a fully nonlinear equation on four-manifolds. J. Anal. Math. 87, 151–186 (2002)

    Google Scholar 

  5. Chou K.-S., Wang X.-J. (2000) A logarithmic Gauss curvature flow and the Minkowski problem. Ann. Inst. Henri Poicaré, Analyse non linéaire, 17, 733–751

    Article  MATH  MathSciNet  Google Scholar 

  6. Chow B. (1997) Geometric aspects of Aleksandrov reflection and gradient estimates for parabolic equations. Comm. Anal. Geom. 5, 389–409

    MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  8. Guan P., Wang G. (2003) A fully nonlinear conformally flow on locally conformally flat manifold. J. Reine Angew. Math. 557, 219–238

    MATH  MathSciNet  Google Scholar 

  9. Gursky M., Viaclovsky J. (2003) Fully nonlinear equations on Riemannian manifolds with negative curvature. Indiana Univ. Math. J. 52, 399–419

    Article  MATH  MathSciNet  Google Scholar 

  10. Krylov, N. Nonlinear elliptic and parabolic equations of the second order. D. Reidel, (1987)

  11. Lee J., Parker T. (1987) The Yamabe problem. Bull. Amer. Math. Soc. 17, 37–91

    Article  MATH  MathSciNet  Google Scholar 

  12. Li A., Li Y.-Y. (2003) On some conformally invariant fully nonlinear equations. Comm. Pure. Appl. Math. 56, 1416–1464

    Article  MATH  MathSciNet  Google Scholar 

  13. Li J., Sheng W.M. (2005) Deforming metrics with negative curvature by a fully nonlinear flow. Calc. Var. PDE. 23, 33–50

    Article  MATH  MathSciNet  Google Scholar 

  14. Lohkamp, J. Metrics of negative Ricci curvature. Ann. of Math. (2), 140, 655–683 (1994)

    Google Scholar 

  15. Viaclovsky J. (2000) Conformal geometry, contact geometry and the calculus of variations. Duke Math. J. 101, 283–316

    Article  MATH  MathSciNet  Google Scholar 

  16. Viaclovsky J. (2002) Estimates and some existence results for some fully nonlinear elliptic equations on Riemannian manifolds. Comm. Analysis and Geometry 10, 815–846

    MATH  MathSciNet  Google Scholar 

  17. Ye R. (1994) Global existencs and convergence of Yamabe flow. J. Differ. Geom. 39, 35–50

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Zhejiang University, Hangzhou, 310028, China

    Weimin Sheng

  2. Department of Physics, Zhejiang University, Hangzhou, 310027, China

    Yan Zhang

Authors
  1. Weimin Sheng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yan Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Weimin Sheng.

Additional information

The authors were supported by NSFC grant number 10471122.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Sheng, W., Zhang, Y. A class of fully nonlinear equations arising from conformal geometry. Math. Z. 255, 17–34 (2007). https://doi.org/10.1007/s00209-006-0011-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-006-0011-5

Keywords

Search

Navigation