Skip to main content
SpringerLink
Log in

Deforming metrics with negative curvature by a fully nonlinear flow

  • Published:
Calculus of Variations and Partial Differential Equations Aims and scope Submit manuscript

Abstract.

By studying a fully nonlinear flow deforming conformal metrics on compact and connected manifold, we prove that for \(\lambda < 1\), any metric g with its modified Schouten tensor \(A^\lambda_{g}\in \Gamma_k^-\) always can be deformed in a natural way to a conformal metric with constant \(\sigma_k\)-scalar curvature at exponential rate.

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.

Similar content being viewed by others

References

  1. Andrew, B.: Monotone quantities and unique limits for evolving convex hypersurfaces. Internat. Math. Res. Notices 1997, 1001-1031 (1997)

    Article  Google Scholar 

  2. Brendle, S., Viaclovsky, J.: A variational characterization for \(\sigma_{n/2}\). Calc. Var. (to appear)

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

    Google Scholar 

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

    Google Scholar 

  5. 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 

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

    Google Scholar 

  7. Chou, K.-S., Wang, X.-J.: A variational theory of the Hessian equation. Comm. Pure Appl. Math. 54, 1029-1064 (2001)

    Article  Google Scholar 

  8. Chow, B.: The Yamabe flow on locally conformally flat manifolds with positive Ricci curvature. Comm. Pure Appl. Math. 45, 1003-1014 (1992)

    Google Scholar 

  9. Guan, P., Viaclovsky, J., Wang, G.: Some properties of the Schouten tensor and applications to conformal geometry. Trans. AMS. 355, 925-933 (2003)

    Article  Google Scholar 

  10. Guan, P., Wang, G.: Lccal estimates for a class of conformal equation arising from conformal geometry. Int. Math. Res. Not. 2003, 1413-1432 (2003)

    Article  Google Scholar 

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

    Google Scholar 

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

    Article  Google Scholar 

  13. Hamilton, R.: The Ricci flow on surfaces. Mathematics and general relativity, Contemporary Math. AMS, Vol. 71, pp. 237-262 (1988)

  14. Krylov, N.: Nonlinear elliptic and parabolic equations of the second order. D. Reidel, Publishing Company 1987

  15. Lee, J., Parker, T.: The Yamabe problem. Bull. Amer. Math. Soc. 17, 37-91 (1987)

    Google Scholar 

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

    Article  Google Scholar 

  17. Li, Y.-Y.: Some existence results for fully nonlinear elliptic equations of Monge-Ampére type. Comm. Pure Appl. Math. 43, 233-271 (1990)

    Google Scholar 

  18. Lin, M., Trudinger, N.: On some inequalities for elementary symmetric functions. Bull. Austral. Math. Soc. 50, 317-326 (1994)

    Google Scholar 

  19. Lohkamp, J.: Metrics of negative Ricci curvature. Ann. of Math. 140(2), 655-683 (1994)

    Google Scholar 

  20. Simon, L.: Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems. Ann. of Math. 118, 525-571 (1983)

    Google Scholar 

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

    Article  Google Scholar 

  22. Viaclovsky, J.: Some fully nonlinear equations in conformal geometry. In: Differential equations and mathematical physics (Birmingham, AL, 1999), pp 425-433. Amer. Math. Soc., Providence, RI 2000

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

    Google Scholar 

  24. Ye, R.: Global existencs and convergence of Yamabe flow. J. Differential Geometry 39, 35-50 (1994)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics, Chinese Academy of Sciences, 100080, Beijing, China

    Jiayu Li & Weimin Sheng

Authors
  1. Jiayu Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Weimin Sheng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jiayu Li.

Additional information

Received: 2 December 2003, Accepted: 10 May 2004, Published online: 16 July 2004

Jiayu Li: Supported in part by a grant from the National Science Foundation of China.

Weimin Sheng: Supported by the Zhejiang Provincial Natural Science Foundation of China (No.102033).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Li, J., Sheng, W. Deforming metrics with negative curvature by a fully nonlinear flow. Calc. Var. 23, 33–50 (2005). https://doi.org/10.1007/s00526-004-0287-4

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00526-004-0287-4

Keywords

Search

Navigation