The Journal of Geometric Analysis

, Volume 29, Issue 1, pp 510–541 | Cite as

Geometry and Topology of the Space of Plurisubharmonic Functions

  • Soufian AbjaEmail author


Let \(\Omega \) be a strongly pseudoconvex domain. We introduce the Mabuchi space of strongly plurisubharmonic functions in \(\Omega \). We study the metric properties of this space using Mabuchi geodesics and establish regularity properties of the latter, especially in the ball. As an application, we study the existence of local Kähler–Einstein metrics.


Geodesics Mabuchi space Monge–Ampère equation Pseudoconvex domain 

Mathematics Subject Classification

53C55 32W20 32U05 



The author is grateful for his supervisors Vincent Guedj and Said Asserda, for their support, suggestions and encouragement. The author wants to thank Ahmed Zeriahi for his very useful discussions and suggestions. The author would also like to thank Tat Dat Tô and Zakarias Sjöström Dyrefelt for their very careful reading of the preliminary version of this paper and their very useful discussions. This work has been finalized while the author was visiting the “Institut de Mathématiques de Toulouse” in march 2017 and he would like to thank them for their hospitality.


  1. 1.
    Bedford, E., Taylor, B.A.: The Dirichlet problem for a complex Monge–Ampère equation. Invent. Math. 37(1), 1–44 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bedford, E., Taylor, B.A.: A new capacity for plurisubharmonic functions. Acta Math. 149(1–2), 1–40 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Berman, R., Berndtsson, B.: Moser-Trudinger type inequalities for complex Monge-Ampère operators and Aubin’s hypothèse fondamentale, ArXiv e-prints, arXiv:1109.1263B (2011)
  4. 4.
    Berman, R., Boucksom, S., Guedj, V., Zeriahi, A.: A variational approach to complex Monge–Ampère equations. Publ. Math. I.H.E.S. 117, 179–245 (2013)CrossRefzbMATHGoogle Scholar
  5. 5.
    Berndtsson, B.: Subharmonicity properties of the Bergman kernel and some other functions associated to pseudoconvex domains. Annales de l’institut Fourier. vol. 56(6) (2006)Google Scholar
  6. 6.
    Berndtsson, B.: A Brunn–Minkowski type inequality for Fano manifolds and some uniqueness theorems in Kähler geometry. Inv. Math. 200(1), 149–200 (2015)CrossRefzbMATHGoogle Scholar
  7. 7.
    Calabi, E., Chen, X. X.: The space of Kähler metrics. II. J.D.G.. In: Measures of finite pluricomplex energy pp. 173–193(2002)Google Scholar
  8. 8.
    Cegrell, U.: Pluricomplex energy. Acta Math. 180, 187–217 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Cegrell, U.: Measures of finite pluricomlex energy. arXiv:1107.1899
  10. 10.
    Chen, X.X.: The space of Kähler metrics. J. Diff. Geom. 56, 189–234 (2000)CrossRefzbMATHGoogle Scholar
  11. 11.
    Chen, X.X.: Space of Kähler metrics III. On the lower bound of the Calabi energy and geodesic distance. Invent. Math. 175(n3), 453–503 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Chen, X.X., Tian, G.: Geometry of Kähler metrics and foliations by holomorphic discs. Publ. Math. Inst. Hautes Etudes Sci. 107, 1–107 (2008)CrossRefzbMATHGoogle Scholar
  13. 13.
    Darvas, T.: Weak Geodesic Rays in the Space of Kähler Metrics and the Class \({{\cal{E}}}(X,\omega )\). J. Inst. Math. Jussieu 16(4), 837–858 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Darvas, T.: The Mabuchi completion of the space of Khler potentials. Am. J. Math. 139(5), 1275–1313 (2017)CrossRefzbMATHGoogle Scholar
  15. 15.
    Darvas, T.: Mabuchi geometry of finite energy classes. Adv. Math. 285, 182–219 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    DiNezza, E., Guedj, V.: Geometry and Topology of the space of Kähler metrics on singular varieties. To appear in CompositioGoogle Scholar
  17. 17.
    Darvas, T., Lempert, L.: Weak geodesics in the space of Kähler metrics. Math. Res. Lett. 19(5), 1127–1135 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Darvas, T., Rubinstein, Y.: Tian’s properness conjectures and Finsler geometry of the space of Kahler metrics. J. Am. Math. Soc. 30(2), 347–387 (2017)CrossRefzbMATHGoogle Scholar
  19. 19.
    Demailly, J. -P.: Monge–Ampère operators, Lelong numbers and intersection theory. In: Complex analysis and geometry, pp. 15–193, University Series in Mathematics Plenum, New York (1993)Google Scholar
  20. 20.
    Donaldson, S. K.: Symmetric spaces, Kähler geometry and Hamiltonian dynamics. vol. 196 of A.M.S. Transl. Ser. 2, pp. 13–33. American Mathematical Society, Providence (1999)Google Scholar
  21. 21.
    Guedj, V.: The metric completion of the Riemannian space of Kähler metrics. arXiv:1401.7857
  22. 22.
    Guedj, V., Kolev, B., Yeganefar, N.: Kähler–Einstein fillings. J. Lond. Math. Soc. 88(3), 737–760 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Guedj, V., Zeriahi, A.: Degenerate complex Monge–Ampère equations. In: EMS Tracts in Mathematics, 26. European Mathematical Society (EMS), Zrich (2017)Google Scholar
  24. 24.
    Hosono, G.: Local geodesics between toric plurisubharmonic functions with infinite energy. Ann. Polon. Math. 120, 33–40 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Kolev, B.: The riemannian space of K ähler. In: V. Guedj (ed) Complex Monge–Ampère equations and geodesics in the space of Kähler metrics. Lecture Notes in Mathematics, vol. 2038, Springer (2012)Google Scholar
  26. 26.
    Lempert, L., Vivas, L.: Geodesics in the space of Kähler metrics. Duke Math. J. 162(7), 1369–1381 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Mabuchi, T.: Some symplectic geometry on compact Kähler manifolds. I. Osaka J. Math. 24(2), 227–252 (1987)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Rashovskii, A.: Local geodesics for plurisubharmonic functions. Math. Z. 287(1), 73–83 (2017)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Semmes, S.: Complex Monge–Ampère and symplectic manifolds. Am. J. Math. 114(3), 495–550 (1992)CrossRefzbMATHGoogle Scholar
  30. 30.
    Zeriahi, A.: Volume and capacity of sublevel sets of a Lelong class of plurisubharmonic functions. Indiana Univ. Math. J. 50(1), 671–703 (2001)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 2018

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of SciencesIbn Tofail UniversityKenitraMorocco

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