[Paper Review] Dynamical construction of Kähler-Einstein metrics on bounded pseudoconvex domains

Hajime Tsuji|arXiv (Cornell University)|Nov 16, 2013

Geometry and complex manifolds6 references3 citations

TL;DR

This paper establishes a dynamical construction of Kähler-Einstein metrics on bounded strongly pseudoconvex domains in ℂⁿ with C⁰-boundary by showing that the normalized limit of a sequence of Bergman kernels—defined via iterated L²-orthonormal bases with respect to weighted metrics—converges to the Kähler-Einstein volume form. The key result is that limₘ→∞ √[m]{(m!)⁻ⁿKₘ} = (2π)⁻ⁿdV_E, where dV_E is the Kähler-Einstein volume form.

ABSTRACT

In this note we shall prove that the complete Kähler-Einstein volume form on a bounded strongly pseudoconvex domain with $C^{\infty}$-boundary is the normalized limit of a sequence of Bergman kernels.

Motivation & Objective

• To establish a dynamical system based on Bergman kernels that converges to the complete Kähler-Einstein metric on bounded strongly pseudoconvex domains with C⁰-boundary.
• To extend the noncompact version of the result from [T, p.110, Theorem 1.2] to domains with smooth boundary, ensuring uniform estimates.
• To demonstrate that the normalized m-th root of the m-th Bergman kernel converges to the Kähler-Einstein volume form in the compact uniform topology.
• To generalize the construction to general bounded pseudoconvex domains via exhaustion by strongly pseudoconvex subdomains.
• To prove semipositivity of the relative Kähler-Einstein volume form in families using curvature estimates on the limit metric.

Proposed method

• Define a recursive dynamical system: K₁ = Bergman kernel of Ω, h₁ = K₁⁻¹, and recursively Kₘ₊₁ = Bergman kernel of H⁰(Ω, (m+1)Kₐ) with respect to the L²-inner product weighted by hₘ.
• Use Hörmander’s L²-estimate for ∂̄ to ensure that the spaces A²(Ω, (m+1)Kₐ, hₘ) are well-defined and very ample, enabling the construction of Kₘ₊₁.
• Leverage the bounded geometry of (Ω, ω_E) of infinite order (from Cheng-Yau) to control curvature and metric decay uniformly across Ω.
• Establish upper and lower L²-estimates for sections using Taylor expansion of the metric determinant and comparison with the Euclidean model.
• Apply Yau’s Schwarz lemma to show monotonicity of the volume forms ω_E,cⁿ on sublevel sets Ω_c, enabling the construction of the limit metric.
• Use Berndtsson’s theorem on curvature positivity of direct images to prove semipositivity of the relative volume form dV_Ω/Δ⁻¹ on families of domains.

Experimental results

Research questions

• RQ1 Can the complete Kähler-Einstein metric on a bounded strongly pseudoconvex domain with C⁰-boundary be realized as the limit of a sequence of Bergman metrics?
• RQ2 Does the normalized m-th root of the m-th Bergman kernel converge uniformly to the Kähler-Einstein volume form?
• RQ3 How can uniform estimates be maintained in the noncompact setting of strongly pseudoconvex domains?
• RQ4 Is the relative Kähler-Einstein volume form semipositive in families of pseudoconvex domains?
• RQ5 Can the construction of the Kähler-Einstein metric be extended from strongly pseudoconvex to general bounded pseudoconvex domains via exhaustion?

Key findings

• The normalized limit limₘ→∞ √[m]{(m!)⁻ⁿKₘ} exists in the compact uniform topology on Ω.
• This limit equals (2π)⁻ⁿdV_E, where dV_E is the Kähler-Einstein volume form associated with the complete Kähler-Einstein metric ω_E.
• The convergence is uniform and holds pointwise on the entire domain Ω, with the limit independent of the choice of orthonormal basis.
• The proof relies on uniform L²-estimates derived from the bounded geometry of (Ω, ω_E), which ensures control over curvature and metric decay.
• The construction extends to general bounded pseudoconvex domains via exhaustion by strongly pseudoconvex subdomains Ω_c, where the Kähler-Einstein metric is constructed as a monotone decreasing limit.
• The relative volume form dV_Ω/Δ⁻¹ on Ω ⊂ ℂⁿ×Δ has semipositive curvature, proven via Berndtsson’s theorem and the limit construction.