Abstract
We study the complete Kähler-Einstein metric of a Hartogs domain \(\tilde \Omega \) built on an irreducible bounded symmetric domain sW, using a power N µ of the generic norm of Ω. The generating function of the Kähler-Einstein metric satisfies a complex Monge-Ampère equation with a boundary condition. The domain \(\tilde \Omega \) is in general not homogeneous, but it has a subgroup of automorphisms, the orbits of which are parameterized by X ε [0, 1[. This allows us to reduce the Monge-Ampère equation to an ordinary differential equation with a limit condition. This equation can be explicitly solved for a special value µ0 of µ. We work out the details for the two exceptional symmetric domains. The special value µ0 seems also to be significant for the properties of other invariant metrics like the Bergman metric; a conjecture is stated, which is proved for the exceptional domains.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Cheng S Y, Yau S T. On the existence of a complete Kähler metric on non-compact complex manifolds and the regularity of Fefferman’s equation. Comm Pure Appl Math, 1980, 33: 507–544
Mok N, Yau S T. Completeness of the Kähler-Einstein metric on bounded domain and the characterization of domain of holomorphy by curvature conditions. Proc Symposia Pure Math, 1983, 39: 41–59
Wu H. Old and new invariants metrics on complex manifolds. In: Fornaess J E, eds. Several Complex Variables: Proceedings of the Mittag-Leffler Institute, 1987–1988, Math Notes, Vol 38. Princeton: Princeton Univ Press, 1993, 640–682
Yin W P, Lu K P, Roos Guy. New classes of domains with explicit Bergman kernel. Sci China, Ser A, 2004, 47: 352–371
Wang A, Yin W P, Zhang L Y, et al. The Einstein-Kähler metric with explicit formula on nonhomogeneous domain. Asian J Math, 2004, 8: 39–50
Cartan H. Les fonctions de deux variables complexes et le problème de la représentation analytique. J Math Pures Appl, 1931, 10: 1–114
Loos Ottmar. Bounded Symmetric Domains and Jordan Pairs. Math Lectures, Univ of California, Irvine, 1977
Faraut J, Kaneyuki S, Korányi A, et al. Analysis and geometry on complex homogeneous domains. In: Progress in Mathematics. Boston: Birkhäuser, 1999, 425–534
Yin W P, Wang A, Zhao X X. Comparison theorem on Cartan-Hartogs domain of the first type. Sci China, Ser A, 2001, 44(5): 587–598
Hua L K. Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains. Providence: American Mathematical Society, 1963
Loos O, Jordan P. Lecture Notes in Mathematics, Vol 460. Berlin-Heidelberg-New York: Springer-Verlag, 1975
Roos G, Vigué J P. Systèmes triples de Jordan et domaines symétriques. Hermann, Paris, Travaux en cours, 1992, 43: 1–84
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Wang, A., Yin, W., Zhang, L. et al. The Kähler-Einstein metric for some Hartogs domains over symmetric domains. SCI CHINA SER A 49, 1175–1210 (2006). https://doi.org/10.1007/s11425-006-0230-6
Received:
Accepted:
Issue date:
DOI: https://doi.org/10.1007/s11425-006-0230-6