Balanced metrics on some Hartogs type domains over bounded symmetric domains

Abstract

The definition of balanced metrics was originally given by Donaldson in the case of a compact polarized Kähler manifold in 2001, who also established the existence of such metrics on any compact projective Kähler manifold with constant scalar curvature. Currently, the only noncompact manifolds on which balanced metrics are known to exist are homogeneous domains. The generalized Cartan-Hartogs domain k j=1 Ω j B d 0 (µ) is defined as the Hartogs type domain constructed over the product k j=1 Ω j of irreducible bounded symmetric domains Ω j (1 ≤ j ≤ k), with the fiber over each point (z 1 , · · · , z k ) ∈ k j=1 Ω j being a ball in C d 0 of the radius k j=1 N Ω j (z j , z j ) µ j 2 of the product of positive powers of their generic norms. Any such domain k j=1 Ω j B d 0 (µ) (k ≥ 2) is a bounded nonhomogeneous domain. The purpose of this paper is to obtain necessary and sufficient conditions for the metric αg(µ) (α > 0) on the domain k j=1 Ω j B d 0 (µ) to be a balanced metric, where g(µ)is its canonical metric. As the main contribution of this paper, we obtain the existence of balanced metrics for a class of such bounded nonhomogeneous domains.

Kähler Immersions of Kähler Manifolds into Complex Space Forms

“…Cartan-Hartogs domains has been considered by many authors (see e.g. [30,31,50,51,53,76,77,79,80,81,82]) under different points of view. Their importance relies on being examples of nonhomogeneous domains which for a particular value of the parameter µ are Kähler-Einstein.…”

Strongly not relatives Kähler manifolds

“…with K the Bergman kernel of Ω. Consider on M Ω (µ) the metric g(µ) whose associated Kähler form ω(µ) can be described by the (globally defined) Kähler potential centered at the origin (7) Φ(z, w) = − log(K(z, z) −µ − |w| 2 ).…”

Rawnsley’s ε-function on some Hartogs type domains over bounded symmetric domains and its applications

“…For the reference of the generalized Cartan-Hartogs domains, see Feng-Tu [15], Tu-Wang [24] and Wang-Hao [25].…”

“…If ν = 0, then the metric g(µ; ν) becomes the standard canonical metric (e.g., see Bi-Tu [3], Feng-Tu [14,15], Loi-Zedda [20] and Zedda [27,28]). In this paper, we will focus our attention on the metric…”

“…For the generalized Cartan-Hartogs domain k j=1 Ω j B d 0 (µ), g(µ; ν) in the case of ν = 0, Feng-Tu [15] proved that the Rawnsley's ε-function admits the following expansion : [15]). Let Ω i be an irreducible bounded symmetric domain in C d i , and denote the generic norm N Ω i (z i , z i ), the dimension d i and the genus p i for…”