[Paper Review] Kähler geometry of bounded pseudoconvex Hartogs domains

Yihong Hao, An Wang|arXiv (Cornell University)|Nov 17, 2014

Geometry and complex manifolds3 citations

TL;DR

This paper investigates the Kähler geometry of bounded pseudoconvex Hartogs domains by analyzing a natural complete Kähler metric $ g^ olimits $ derived from the domain's defining function. It establishes conditions under which $ g^ olimits $ is Kähler-Einstein or extremal, and determines when the domain admits holomorphic isometric immersions into finite or infinite-dimensional complex space forms, using Calabi's diastasis criterion and curvature analysis of the base domain.

ABSTRACT

Let $Ω$ be a bounded pseudoconvex Hartogs domain. There exists a natural complete Kähler metric $g^Ω$ in terms of its defining function. In this paper, we study two problems. The first one is determining when $g^Ω$ is Einstein or extremal. The second one is the existence of holomorphic isometric immersions of $(Ω, g^Ω)$ into finite or infinite dimensional complex space forms.

Motivation & Objective

• To characterize when the natural Kähler metric $ g^ olimits $ on a bounded pseudoconvex Hartogs domain is Kähler-Einstein or extremal.
• To determine the existence of holomorphic isometric immersions of $ (\Omega, g^\Omega) $ into complex Euclidean, projective, or hyperbolic space forms.
• To relate the geometric properties of the Hartogs domain to those of its base domain $ D $, particularly via curvature and plurisubharmonic functions.
• To extend previous results on Cartan-Hartogs domains to general bounded pseudoconvex Hartogs domains using the diastasis function and Calabi's criterion.

Proposed method

• Define the Kähler metric $ g^\Omega $ using the Kähler potential $ -\log(\varphi(z) - \|z_0\|^2) $, where $ \varphi $ is a strictly plurisubharmonic exhaustion function on the base domain $ D $.
• Apply Calabi's diastasis criterion to analyze the existence of holomorphic isometric immersions into complex space forms by examining the positivity of certain matrices derived from the Taylor expansion of the metric potential.
• Use the condition that $ -\log\varphi $ is strictly plurisubharmonic and $ C^\infty $ to ensure the metric is well-defined and complete.
• Reduce the immersion problem to the base domain by leveraging the circular symmetry of the Hartogs domain, which simplifies the Bochner coordinates of the diastasis.
• Analyze the Ricci curvature and scalar curvature of $ g^\Omega $ to determine when it is Einstein or extremal, using curvature formulas derived from the potential function.
• Employ the continuity method and Monge-Ampère equation techniques, as in Cheng-Yau and Mok-Yau, to relate the natural metric $ g^\Omega $ to the Bergman metric and Kähler-Einstein structures.

Experimental results

Research questions

• RQ1 Under what conditions on the base domain $ D $ is the natural Kähler metric $ g^\Omega $ on a bounded pseudoconvex Hartogs domain $ \Omega $ Kähler-Einstein?
• RQ2 When is the natural Kähler metric $ g^\Omega $ extremal, and how does this relate to the scalar curvature and Ricci curvature of $ g^\Omega $?
• RQ3 When does the Hartogs domain $ (\Omega, g^\Omega) $ admit a holomorphic isometric immersion into complex Euclidean, projective, or hyperbolic space forms?
• RQ4 How does the immersion property of $ (\Omega, g^\Omega) $ relate to the immersion properties of the base domain $ (D, g^D) $?
• RQ5 What role does the parameter $ h $, related to the homothety of the metric, play in determining the existence of immersions into complex hyperbolic space?

Key findings

• The natural Kähler metric $ g^\Omega $ on a bounded pseudoconvex Hartogs domain is Kähler-Einstein if and only if the base metric $ g^D $ has Ricci curvature $ -(d+1) $, where $ d $ is the complex dimension of the base.
• The metric $ g^\Omega $ is extremal if and only if its scalar curvature is constant, which holds precisely when $ g^D $ is Kähler-Einstein with Ricci curvature $ -(d+1) $.
• There exists a holomorphic isometric immersion of $ (\Omega, g^\Omega) $ into $ \mathbb{CH}^\infty $ if and only if the base $ (D, g^D) $ admits such an immersion into $ \mathbb{CH}^N $ for $ N \leq \infty $, under the condition $ 0 < h < 1 $.
• For $ h \geq 1 $, the matrix $ C_{z_{01}(i)}(0) $ fails to be non-negative definite, so no such immersion exists into $ \mathbb{CH}^\infty $, and hence not into any finite-dimensional complex hyperbolic space.
• The domain $ (\Omega, g^\Omega) $ does not admit a holomorphic isometric immersion into any finite-dimensional complex space form, as shown by the non-existence of positive definite matrices in the diastasis expansion.
• The existence of immersions into $ \mathbb{C}^\infty $ and $ \mathbb{CP}^\infty $ depends on the base domain’s immersion properties and the parameter $ h $, with $ \mathbb{CH}^\infty $ being possible only for $ 0 < h < 1 $.