[Paper Review] Bergman-Einstein metric on a Stein space with a strongly pseudoconvex boundary

Jimmy Xiangji Huang, Xiaoshan Li|arXiv (Cornell University)|Aug 9, 2020

Holomorphic and Operator Theory16 references4 citations

TL;DR

This paper proves that if the Bergman metric on the regular part of a Stein space with a compact, strongly pseudoconvex boundary is Kähler-Einstein, then the boundary is spherical. The result generalizes the Cheng conjecture to singular complex spaces using Bergman kernel theory and Kähler-Einstein metric analysis on Stein spaces with isolated singularities.

ABSTRACT

Let $Ω$ be a Stein space with a compact smooth strongly pseudoconvex boundary. We prove that the boundary is spherical if its Bergman metric over $\hbox{Reg}(Ω)$ is Kähler-Einstein.

Motivation & Objective

• To generalize the Cheng conjecture on the equivalence of Bergman and Kähler-Einstein metrics to Stein spaces with isolated singularities.
• To investigate the geometric implications of the Bergman metric being Kähler-Einstein on the regular part of a Stein space with compact, strongly pseudoconvex boundary.
• To establish a characterization of the boundary as spherical under the Kähler-Einstein condition on the Bergman metric.
• To extend results from smooth domains to singular complex spaces using Bergman kernel theory and pluripotential methods.
• To provide a partial solution to the generalized Cheng conjecture in the context of normal Stein spaces with spherical boundaries.

Proposed method

• Define the Bergman space $ A^2(ar{ abla}) $ on the regular part of a Stein space $ abla $ with compact, strongly pseudoconvex boundary.
• Construct the Bergman kernel form $ K_{ abla} = \sum f_j \wedge \overline{f_j} $ using an orthonormal basis of holomorphic $ (n,0) $-forms.
• Define the Bergman metric $ \omega_{\nabla}^B = i\partial\overline{\partial}\log k_{\nabla}(z,\overline{z}) $ on the regular locus $ \mathrm{Reg}(\nabla) $.
• Use the existence of a resolution $ \pi: M \to \nabla $ with exceptional divisor $ E $, where $ M $ is a strongly pseudoconvex manifold.
• Apply Ohsawa-Takegoshi and Hörmander $ L^2 $-estimates to extend $ L^2 $-holomorphic $ (n,0) $-forms from $ M $ to a larger manifold $ M' $.
• Analyze the asymptotic behavior of the Bergman kernel near the boundary and use the Monge-Ampère equation $ \det(u_{i\overline{j}}) = c e^u $ to deduce pluriharmonic properties.

Experimental results

Research questions

• RQ1 Under what conditions does the Bergman metric on a singular Stein space with strongly pseudoconvex boundary become Kähler-Einstein?
• RQ2 Does the Kähler-Einstein property of the Bergman metric imply that the boundary is spherical?
• RQ3 Can the generalized Cheng conjecture be extended to normal Stein spaces with isolated singularities?
• RQ4 What is the role of the Bergman kernel's vanishing order at singular points in determining the metric's Einstein property?
• RQ5 How does the uniformization theorem and the uniqueness of the Cheng-Yau metric constrain the possible group actions on the ball in the singular quotient case?

Key findings

• If the Bergman metric on the regular part of a Stein space $ \Omega $ with compact, strongly pseudoconvex boundary is Kähler-Einstein, then the boundary $ \partial\Omega $ is spherical.
• The proof relies on the asymptotic expansion of the Bergman kernel near the boundary and the solution of the Monge-Ampère equation $ \det(u_{i\overline{j}}) = c e^u $ with $ u = \log K_\Gamma $.
• For finite group quotients $ \mathbb{B}^n / \Gamma $ with $ n \geq 2 $, if the Bergman kernel vanishes at the origin, the metric cannot be Kähler-Einstein.
• In dimension $ n = 1 $, the Bergman metric on $ \mathbb{B}^1 / \Gamma $ is Kähler-Einstein for any finite $ \Gamma $, showing a dimension-dependent dichotomy.
• The condition $ K_\Gamma(0,0) = 0 $ in $ \mathbb{B}^3 / \mathbb{Z}_2 $ implies the Bergman metric fails to satisfy the Monge-Ampère equation required for Kähler-Einstein structure.
• The uniqueness of the Cheng-Yau Kähler-Einstein metric on the ball implies that only trivial quotients ($ \Gamma = \{\mathrm{id}\} $) yield Kähler-Einstein Bergman metrics in higher dimensions.