Skip to main content
SpringerLink
Log in

A maximum rank problem for degenerate elliptic fully nonlinear equations

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Abstract

The solutions to the Dirichlet problem for two degenerate elliptic fully nonlinear equations in n + 1 dimensions, namely the real Monge–Ampère equation and the Donaldson equation, are shown to have maximum rank in the space variables when n ≤ 2. A constant rank property is also established for the Donaldson equation when n = 3.

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. Alvarez O., Lasry J.M., Lions P.L.: Convexity viscosity solutions and state constraints. J. Math. Pure Appl. 76, 265–288 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bian B., Guan P.: A microscopic convexity principle for nonlinear partial differential equations. Invent. Math. 177, 307–335 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bian B., Guan P.: A structural condition for microscopic convexity principle. Discret. Contin. Dyn. Syst. 28, 789–807 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. Blocki, Z.: On geodesics in the space of Kähler metrics. In: The Proceedings of the “Conference on Geometry” in honor of Shing-Tung Yau, Warsaw (2009)

  5. Brascamp H.J., Lieb E.H.: On extensions of the Brunn–Minkowski and Prokopa–Leindler theorems, including inequalities for log-concave functions with an application to the diffusion equation. J. Funct. Anal. 22, 366–389 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  6. Caffarelli L., Friedman A.: “Convexity of solutions of some semilinear elliptic equations” Duke Math. Duke Math. J. 52, 431–455 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  7. Caffarelli L., Guan P., Ma X.: A constant rank theorem for solutions of fully nonlinear elliptic equations. Commun. Pure Appl. Math. 60, 1769–1791 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  8. Caffarelli L., Nirenberg L., Spruck J.: The Dirichlet problem for non-linear second-order elliptic equations I. Monge–Ampère equations. Commun. Pure Appl. Math. XXXVII, 369–402 (1984)

    Article  MathSciNet  Google Scholar 

  9. Chen X.X.: The space of Kähler metrics. J. Differ. Geom. 56, 189–234 (2000)

    MATH  Google Scholar 

  10. Chen, X.X., He, W.: The space of volume forms. arXiv: 0810.3880

  11. Chen, X.X., Sun, S.: Space of Kähler metrics (V): Kähler quantization. arXiv: 0902.4149

  12. Donaldson S.K.: Symmetric spaces, Kähler geometry, and Hamiltonian dynamics. Am. Math. Soc. Transl. 196, 13–33 (1999)

    MathSciNet  Google Scholar 

  13. Donaldson S.K.: Nahm’s equations and free boundary problems. In: The many facets of Geometry, pp 71–91. Oxford University Press, Oxford (2010)

  14. Friedman A.: Partial Differential Equations of Parabolic Type. Prentice-Hall, Englewood Cliffs (1964)

    MATH  Google Scholar 

  15. Guan B.: The Dirichlet problem for Hessian equations on Riemannian manifolds. Calc. Var. Partial Differ. Equ. 8, 45–69 (1999)

    Article  MATH  Google Scholar 

  16. Guan D.: On modified Mabuchi functional and Mabuchi moduli space of Kähler metrics on toric bundles. Math. Res. Lett. 6(5–6), 547–555 (1999)

    MathSciNet  MATH  Google Scholar 

  17. Guan P., Li Q., Zhang X.: A uniqueness theorem in Kähler geometry. Math. Ann. 345, 377–393 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  18. He, W.: The Donaldson equation. arXiv: 0810.4123

  19. Korevaar N.J., Lewis J.: Convex solutions of certain elliptic equations have constant rank Hessians. Arch. Rat. Mech. Anal. 97, 19–32 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  20. Lempert L.: La métrique de Kobayashi et la représentation des domaines sur la boule. Bull. Soc. Math. France 109, 427–474 (1981)

    MathSciNet  MATH  Google Scholar 

  21. Li Q.: Constant rank theorem in complex variables. Indiana Univ. Math. J. 58, 1235–1256 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  22. Mabuchi T.: Some symplectic geometry on compact Kähler manifolds. Osaka J. Math. 24, 227–252 (1987)

    MathSciNet  MATH  Google Scholar 

  23. Phong D.H., Sturm J.: The Monge–Ampère operator and geodesics in the space of Kähler potentials. Invent. Math. 166, 125–149 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  24. Phong D.H., Sturm J.: The Dirichlet problem for degenerate complex Monge–Ampère equations. Commun. Anal. Geometry 18, 145–170 (2010)

    MathSciNet  MATH  Google Scholar 

  25. Phong D.H., Sturm J.: Test configurations for K-stability and geodesic rays. J. Symplectic Geom. 5(2), 221–247 (2007)

    MathSciNet  MATH  Google Scholar 

  26. Semmes S.: Complex Monge–Ampère equations and symplectic manifolds. Am. J. Math. 114, 495–550 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  27. Singer I.M., Wong B., Yau S.S.T., Yau S.T.: An estimate of gap of the first two eigenvalues of the Schrödinger operator. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 12(4), 319–333 (1985)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, McGill University, Montreal, QC, H3A 2K6, Canada

    Pengfei Guan

  2. Department of Mathematics, Columbia University, New York, NY, 10027, USA

    D. H. Phong

Authors
  1. Pengfei Guan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. D. H. Phong
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to D. H. Phong.

Additional information

Research of P. Guan was supported in part by NSERC Discovery Grant. Research of the D. H. Phong was supported in part by the National Science Foundation grant DMS-07-57372.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Guan, P., Phong, D.H. A maximum rank problem for degenerate elliptic fully nonlinear equations. Math. Ann. 354, 147–169 (2012). https://doi.org/10.1007/s00208-011-0729-1

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-011-0729-1

Keywords

Search

Navigation