Skip to main content
SpringerLink
Log in

Weak solutions to degenerate complex Monge–Ampère flows I

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Abstract

Studying the (long-term) behavior of the Kähler-Ricci flow on mildly singular varieties, one is naturally lead to study weak solutions of degenerate parabolic complex Monge–Ampère equations. The purpose of this article, the first of a series on this subject, is to develop a viscosity theory for degenerate complex Monge–Ampère flows in domains of \({\mathbb {C}}^n\).

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. Alexandrov, A.D.: Almost everywhere existence of the second order differential of a convex function and some properties of convex functions. Leningrad. Univ. Ann. (Math. Ser.) 37, 3–35 (1939). (Russian)

    Google Scholar 

  2. Avelin, B., Hed, L., Persson, H.: Approximation and bounded plurisubharmonic exhaustion functions beyond lipschitz domains. arXiv:1210.7105

  3. Azagra, D., Ferrara, D., Sanz, B.: Viscosity solutions to second order partial differential equations on Riemannian manifolds. J. Diff. Equ. 245, 307–336 (2008)

    Article  Google Scholar 

  4. Berman, R.: Bergman kernels for weighted polynomials and weighted equilibrium measures of \(\mathbb{C}^{n}\). Indiana Univ. Math. J. 58(4), 1921–1946 (2009)

    Article  MathSciNet  Google Scholar 

  5. Berman, R., Demailly, J.-P.: Regularity of plurisubharmonic upper enveloppes in big cohomology classes. Perspectives in analysis, geometry, and topology, 3966, Progr. Math., 296, Birkhäuser, Springer, New York (2012)

  6. Bedford, E., Taylor, B.A.: The Dirichlet problem for a complex Monge–Ampère equation. Invent. Math. 37(1), 1–44 (1976)

    Article  MathSciNet  Google Scholar 

  7. Bedford, E., Taylor, B.A.: A new capacity for plurisubharmonic functions. Acta Math. 149, 1–40 (1982)

    Article  MathSciNet  Google Scholar 

  8. Bremermann, H.J.: On a generalized Dirichlet problem for plurisubharmonic functions and pseudo-convex domains. Characterization of ilov boundaries. Trans. Am. Math. Soc. 91, 246–276 (1959)

    MathSciNet  Google Scholar 

  9. Cardaliaguet, P.: Solutions de viscosité d’équations elliptiques et paraboloiques non linéaires. Notes de Cours, Université de Rennes, Janvier (2004)

  10. Cegrell, U.: On the Dirichlet problem for the complex Monge–Ampère operator. Math. Z. 185(2), 247–251 (1984)

    Article  MathSciNet  Google Scholar 

  11. Caffarelli, L.A., Cabré, X.: Fully nonlinear elliptic equations. American Mathematical Society Colloqium publications, vol. 43 (1995)

  12. Crandall, M., Ishii, H., Lions, P.L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. 27, 1–67 (1992)

    Article  MathSciNet  Google Scholar 

  13. Droniou, J., Imbert, C.: Solutions et solutions variationnelles pour EDP non lineaire, Cours polycopié, Université de Montpellier (2004)

  14. Eyssidieux, P., Guedj, V., Zeriahi, A.: Viscosity solutions to degenerate complex Monge–Ampère equations. Comm. Pure Appl. Math. 64(8), 1059–1094 (2011)

    Article  MathSciNet  Google Scholar 

  15. Eyssidieux, P., Guedj, V., Zeriahi, A.: Continuous approximation of quasi-plurisubharmonic functions. arXiv:1311.2866 (2013)

  16. Eyssidieux, P., Guedj, V., Zeriahi, A.: Weak solutions to degenerate complex Monge–Ampère flows II. (2014)

  17. Fornaess, J.-E., Wiegerinck, J.: Approximation of plurisubharmonic functions. Ark. Mat. 27(2), 257–272 (1989)

    Article  MathSciNet  Google Scholar 

  18. Gaveau, B.: Méthodes de contrôle optimal en analyse complexe.I. Résolution d’équations de Monge–Ampère. J. Funct. Anal. 25, 391–411 (1977)

    Article  MathSciNet  Google Scholar 

  19. Guedj, V., Kolodziej, S., Zeriahi, A.: Hölder continuous solutions to Monge–Ampère equations. Bull. London Math. Soc 40, 1070–1080 (2008)

    Article  MathSciNet  Google Scholar 

  20. Harvey, F.R., Lawson, H.B.: Dirichlet duality and the nonlinear Dirichlet problem. Commun. Pure Appl. Math. 62(3), 396–443 (2009)

    Article  MathSciNet  Google Scholar 

  21. Hörmander, L.: Notions of convexity, Progress in Math. Birkhäuser (1994)

  22. Ishii, H.: On uniqueness and existence of viscosity solutions of fully nonlinear second-order elliptic PDEs. Commun. Pure Appl. Math. 42(1), 15–45 (1989)

    Article  Google Scholar 

  23. Ishii, H., Lions, P.L.: Viscosity solutions of fully nonlinear second-order elliptic partial differential equations. J. Diff. Equ. 83, 26–78 (1990)

    Article  MathSciNet  Google Scholar 

  24. Imbert, C., Sylvestre, L.: Introduction to fully nonlinear parabolic equations. An introduction to the Kähler-Ricci flow. Lecture Notes in Math, vol. 2086, Springer, Heidelberg (2013)

  25. Jensen, R.: The maximum principle for viscosity solutions of fully nonlinear second-order partial differential equations. Arch. Rat. Mech. Anal. 101, 1–27 (1988)

    Article  Google Scholar 

  26. Sadullaev, A.: Plurisubharmonic measures and capacities on complex manifolds. (Russian) Uspekhi Mat. Nauk 36((4)(220)), 53–105, 247 (1981)

  27. Sibony, N.: Une classe de domaines pseudoconvexes. Duke Math. J. 55(2), 299–319 (1987)

    Article  MathSciNet  Google Scholar 

  28. Song, J., Tian, G.: Canonical measures and Kähler Ricci flow. J. Am. Math. Soc. 25, 303–353 (2012)

    Article  MathSciNet  Google Scholar 

  29. Song, J., Tian, G.: The Kähler-Ricci flow through singularities. arXiv:0909.4898 (2009)

  30. Wang, Y.: A viscosity approach to the Dirichlet problem for complex Monge–Ampère equations. Math. Z. 272(1–2), 497–513 (2012)

    Article  MathSciNet  Google Scholar 

  31. Zeriahi, A.: A viscosity approach to degenerate complex Monge–Ampère equation. Ann. Fac. Sci. Toulouse Math. (6) 22(4), 843–913 (2013)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgments

We thank Cyril Imbert for useful discussions.

Author information

Authors and Affiliations

  1. Université Joseph Fourier et Institut Universitaire de France, Grenoble, France

    Philippe Eyssidieux

  2. Institut de Mathématiques de Toulouse et Institut Universitaire de France, Université Paul Sabatier, 118 route de Narbonne, 31062, Toulouse cedex 09, France

    Vincent Guedj

  3. Institut de Mathématiques de Toulouse, Université Paul Sabatier, 118 route de Narbonne, 31062, Toulouse cedex 09, France

    Ahmed Zeriahi

Authors
  1. Philippe Eyssidieux
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Vincent Guedj
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ahmed Zeriahi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Vincent Guedj.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Eyssidieux, P., Guedj, V. & Zeriahi, A. Weak solutions to degenerate complex Monge–Ampère flows I. Math. Ann. 362, 931–963 (2015). https://doi.org/10.1007/s00208-014-1141-4

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-014-1141-4

Search

Navigation