Balanced metrics on Cartan and Cartan–Hartogs domains

Abstract

Abstract. This paper consists of two results dealing with balanced metrics (in S. Donaldson terminology) on nonconpact complex manifolds. In the first one we describe all balanced metrics on Cartan domains. In the second one we show that the only Cartan-Hartogs domain which admits a balanced metric is the complex hyperbolic space. By combining these results with those obtained in [13] we also provide the first example of complete, Kähler-Einstein and projectively induced metric g such that αg is not balanced for all α > 0.

Some remarks on the Kähler geometry of the Taub-NUT metrics

“…Consider for instance a Cartan domain of genus γ endowed with its Bergman metric g B . Then αg B is balanced if and only if α > (γ − 1)/γ (see [35,37] for a proof).…”

“…Equations 34 and 35 can be obtained, after a long but straightforward computation, by substituting the previous expressions into (37) and (38).…”

“…The noncompact case Balanced metrics in the noncompact case were defined for the first time in [2] (see also [12,21,25,36,37]) inspired by the geometric quantization theory. Indeed, the metrics αg for which αg is constant play a prominent role in the theory of Berezin quantization by deformation of Kähler manifolds developed in [7][8][9][10].…”

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

“…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…”

Remarks on the canonical metrics on the Cartan–Hartogs domains

“…Recently, Loi-Zedda [11] and Zedda [22] studied the canonical metrics on the CartanHartogs domains. By calculating the scalar curvature k g(µ) , the Laplace ∆k g(µ) of k g(µ) , the norm |R g(µ) | 2 of the curvature tensor R g(µ) and the norm |Ric g(µ) | 2 of the Ricci curvature Ric g(µ) of a Cartan-Hartogs domain (Ω B d 0 (µ), g(µ)), Zedda [22] has proved that if the coefficient a 2 of the Rawnsley's ε-function expansion for the Cartan-Hartogs domain (Ω B (µ), g(µ)) is constant on Ω B (µ), then (Ω B (µ), g(µ)) is Kähler-Einstein.…”