The Kähler-Einstein metric for some Hartogs domains over symmetric domains

Abstract

Abstract. We study the complete Kähler-Einstein metric of a Hartogs domain Ω built on an irreducible bounded symmetric domain Ω, 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 boundary condition. The domain Ω is in general not homogeneous, but it has a subgroup of automorphisms, the orbits of which are parameterized by X ∈ [0, 1[. This allows to reduce the Monge-Ampère equation to an ordinary differential equation with limit condition. This equation can be explicitly solved for a special value µ 0 of µ. We work out the details for the two exceptional symmetric domains. The special value µ 0 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.

Explicit Formulas for the Bergman Kernel on Certain Forelli-Rudin Construction

“…Therefore, we call Ω N1,...,Nr φ1,...,φr Hua construction also. Obviously, when r = 1 the domain Ω N φ is exact the so-called Cartan-Hartogs domain which was studied by many authors [21,19,18]. Moreover, when r = m = 1 the domain is the complex ellipsoid in complex Euclidean space and a further special case r = m = n = N 1 = 1 is the Thullen domain in C 2 .…”

On holomorphic isometric immersions of nonhomogeneous Kähler–Einstein manifolds into the infinite dimensional complex projective space

“…These domains are also called Cartan domains (also see the details in [34,Appendix B] or [2, Section 2]).…”

“…In fact, Loi and Zedda [18] have proved that there do exist such noncompact submanifolds of CP ∞ , namely, Cartan-Hartogs domains (see [35] for the definition of Cartan-Hartogs domain). More precisely, they proved that on the Cartan-Hartogs domains, the explicit Kähler-Einstein metrics obtained by Yin, Roos, the second and the third authors in [35] and [34] are exact projectively induced by using Calabi's criterion (see Theorem 4.2 for the details).…”

On the solution of Dirichlet problem of complex Monge-Ampère equation on Cartan-Hartogs domain of the second type

“…If N j > 1 (j=1, 2, 3, 4), one can not use the same method as above to reduce the Monge-Ampère equation for the Einstein-Kähler metric on Cartan-Hartogs domains Y I (N 1 , m, n; K), Y II (N 2 , n, K), Y III (N 3 , n; K), Y IV (N 4 , n) to the first-order ODE. Paper [12] ( [11] is the Chinese version of [12]) described the Einstein-Kähler metrics on all of the Cartan-Hatogs domains (including two exceptional cases) by using the Jordan triple system, and reducing the Monge-Ampère equation for the metric to the first-order ODE. Further the estimates of holomorphic sectional curvatures under the Einstein-Kähler metrics are also given.…”

“…Therefore the authors of the paper [13,14,18] guessed the volume of the Einstein-Kähler metric (it is also called the generating function of the metric in [3][4][5][6][7][8]10]), and proved that the guess is true, and then got the Einstein-Kähler metric in explicit forms. By using the result in [12], in the paper [17] the authors got the complete Einstein-Kähler metric with the explicit form on Y III (N 3 , q; K) in the case of K = q 2 + 1 q−1 , and the sharp estimate for the holomorphic sectional curvature under this metric, and proved that the complete Einstein-Kähler metric is equivalent to the Bergman metric on Y III (N 3 , q; K) in the case of K = q 2 + 1 q−1 . These are the main results in the paper [13,14,18].…”