Bergman--Einstein Rigidity for Hartogs Domains over Bounded Homogeneous Domains

Abstract: We establish a rigidity theorem for the Bergman metric on Hartogs domains over bounded homogeneous domains. Let $Ω\subset \mathbb C^n$ be a bounded homogeneous domain, let $K_Ω$ denote its Bergman kernel, and consider the Hartogs domain $$ Ω{m,s}:={(z,ζ)\in Ω\times \mathbb C^m:\ |ζ|^2<KΩ(z,\bar z)^{-s}}, \qquad m\ge 1,\quad s>-C_Ω. $$ We prove that, if $s\neq 0$ and the Bergman metric of $Ω{m,s}$ is Kähler--Einstein, then $Ω{m,s}$ is biholomorphic to the unit ball $\mathbb B^{n+m}$. Thus, within this natural family of Hartogs domains, the Einstein condition on the Bergman metric is completely rigid: it characterizes the ball. This may be viewed as a Cheng-type phenomenon beyond the smoothly bounded strictly pseudoconvex setting, for a class of domains built over homogeneous bases. The proof relies on the explicit formula for the Bergman kernel of $Ω_{m,s}$ together with the structural invariants of the base.

• The paper demonstrates that for s ≠ 0, the only Hartogs domain with a Kähler–Einstein Bergman metric is the ball, extending classical rigidity results.
• It reduces the Einstein condition to a scalar ODE and leverages algebraic constraints from the structure polynomial of bounded homogeneous domains.
• The findings provide a diagnostic link between metric properties and domain symmetry, with implications for Kähler–Ricci soliton generalizations.

Bergman–Einstein Rigidity for Hartogs Domains over Bounded Homogeneous Domains

Introduction and Context

The paper "Bergman–Einstein Rigidity for Hartogs Domains over Bounded Homogeneous Domains" (2604.15880) addresses rigidity phenomena for the Bergman metric on a family of Hartogs domains constructed over bounded homogeneous domains in $\mathbb{C}^n$. The research is motivated by classical questions in several complex variables and complex differential geometry: to what extent do curvature properties of canonical Kähler metrics (notably the Bergman metric) constrain the geometry of the underlying domain?

It is classically known, as a consequence of a conjecture by S.-T. Yau and a theorem by S.-Y. Cheng, that for smoothly bounded strictly pseudoconvex domains, the Einstein condition imposed on the Bergman metric is rigid—it forces the domain to be biholomorphic to the complex unit ball. This paper investigates whether similar rigidity persists outside the context of strictly pseudoconvex domains in the setting of Hartogs-type extensions of bounded homogeneous domains.

Hartogs Domains over Homogeneous Bases

Given a bounded homogeneous domain $\Omega \subset \mathbb{C}^n$ with Bergman kernel $K_\Omega$, the Hartogs domain considered is

$$\Omega_{m,s} = \left\{ (z, \zeta) \in \Omega \times \mathbb{C}^m : \|\zeta\|^2 < K_\Omega(z, \bar{z})^{-s} \right\}$$

for $m \in \mathbb{N}^*$ and real parameter $s > -C_\Omega$, where $C_\Omega$ is an explicit structural constant dependent on $\Omega$. These domains interpolating between $\Omega \times \mathbb{B}^m$ ($s=0$) and strictly pseudoconvex Hartogs-type domains have been systematically studied in complex geometry, automorphism group theory, and explicit metric geometry.

Main Rigidity Theorem

The central result established is the following: If $\Omega \subset \mathbb{C}^n$0, the Bergman metric of $\Omega \subset \mathbb{C}^n$1 is Kähler–Einstein if and only if $\Omega \subset \mathbb{C}^n$2 is biholomorphic to the ball $\Omega \subset \mathbb{C}^n$3. Formally,

If $\Omega \subset \mathbb{C}^n$4 and $\Omega \subset \mathbb{C}^n$5 is Kähler–Einstein, then $\Omega \subset \mathbb{C}^n$6.

Thus, the Einstein condition on the Bergman metric is completely rigid in this natural family: among all possible Hartogs domains of this form over bounded homogeneous bases, only the ball allows for a Kähler–Einstein Bergman metric (up to biholomorphism) when $\Omega \subset \mathbb{C}^n$7.

A corollary of this is a characterization of homogeneity: $\Omega \subset \mathbb{C}^n$8 is homogeneous if and only if it is biholomorphic to the ball.

The $\Omega \subset \mathbb{C}^n$9 case is excluded since then $K_\Omega$0, and the metric is a product of Kähler–Einstein metrics, which in general does not result in a biholomorphism with the ball.

Methodology and Rigidity Mechanism

The proof strategy hinges on several technical components:

1. Explicit Bergman Kernel Formula: The analysis exploits an explicit formula for $K_\Omega$1 due to Ishi, Park, and Yamamori, which is expressible in terms of the Bergman kernel of $K_\Omega$2 and certain structural invariants.
2. Reduction to a Scalar ODE: By leveraging the radial symmetry in the fiber variables $K_\Omega$3 and the invariance properties of the Bergman kernel, the Kähler–Einstein equation for the Bergman metric is reduced to a scalar ODE involving a single radial variable.
3. Algebraic Constraints: Through the structure theory of bounded homogeneous domains, specifically the combinatorics of the structure polynomial $K_\Omega$4 encoding intrinsic invariants, the analysis demonstrates that the only way the Einstein condition can be satisfied is if the underlying base is itself biholomorphic to the unit ball and the parameter $K_\Omega$5 is forced to the precise value $K_\Omega$6.
4. Uniqueness and Biholomorphic Rigidity: The confluence of the above points leads to the conclusion that only the ball admits a Kähler–Einstein Bergman metric among all such Hartogs domains. The uniqueness arises from the rigidity in the zeros and degrees of the structure polynomial associated to $K_\Omega$7.

Strong Structural and Contradictory Claims

• Absolute rigidity for $K_\Omega$8: For any nonzero $K_\Omega$9, the only possible domain within this family admitting a Kähler–Einstein Bergman metric is the ball, sharply contradicting expectations suggested by the broader class of homogeneous or nearly homogeneous domains.
• Exclusion of all domains except the ball: The result invalidates the possibility of non-ball Hartogs domains over non-ball homogeneous bases having Kähler–Einstein Bergman metrics; the structure of the domain is completely determined by the Einstein condition in a manner as rigid as the classical strictly pseudoconvex case.

Implications and Connections

Theoretical Implications

This result extends the paradigm of Bergman–Einstein rigidity beyond the setting of smooth strictly pseudoconvex domains to a broader category including non-smooth and more generally constructed Hartogs-type domains. The proof highlights the deep interaction between complex geometric and algebraic invariants (originating in the structure theory of bounded homogeneous domains).

The paper also motivates and formulates an analogous conjecture for Kähler–Ricci solitons, suggesting that any Hartogs domain $$\Omega_{m,s} = \left\{ (z, \zeta) \in \Omega \times \mathbb{C}^m : \|\zeta\|^2 < K_\Omega(z, \bar{z})^{-s} \right\}$$0 whose Bergman metric is a Kähler–Ricci soliton (not only Einstein) must again be biholomorphic to the ball. Evidence from recent results in the literature supports the conjecture that solitonic conditions induce comparable rigidity.

Practical and Future Directions

• Automatic Ball Characterization: The result provides a clear diagnostic criterion for the classification of Hartogs domains with prescribed canonical metric properties, which is valuable for both metric and function-theoretic investigations in several complex variables.
• Kähler–Einstein Metrics in Complex Geometry: Understanding which non-symmetric or nonhomogeneous domains can support canonical metrics is critical for embedding problems, isometry extension results, and function theory on complex manifolds.
• Solitonic Rigidity: Extending the methods to characterize Kähler–Ricci soliton rigidity for even broader classes of complex domains is a promising future direction. Further, the techniques may inform approaches to open problems on the uniqueness and existence of canonical metrics in various moduli settings.
• Interplay with Automorphism Groups: The result elucidates the connection between the existence of canonical (Bergman–Einstein) metrics and the symmetry properties of the domain, potentially informing automorphism group theory and, more generally, the classification of complex geometric structures.

Conclusion

This work establishes a definitive rigidity theorem for Bergman–Einstein metrics on Hartogs domains $$\Omega_{m,s} = \left\{ (z, \zeta) \in \Omega \times \mathbb{C}^m : \|\zeta\|^2 < K_\Omega(z, \bar{z})^{-s} \right\}$$1 over bounded homogeneous domains: among all such domains with $$\Omega_{m,s} = \left\{ (z, \zeta) \in \Omega \times \mathbb{C}^m : \|\zeta\|^2 < K_\Omega(z, \bar{z})^{-s} \right\}$$2, only the ball $$\Omega_{m,s} = \left\{ (z, \zeta) \in \Omega \times \mathbb{C}^m : \|\zeta\|^2 < K_\Omega(z, \bar{z})^{-s} \right\}$$3 can admit a Kähler–Einstein Bergman metric. The explicit reduction to algebraic and one-variable analytic constraints showcases the complete restriction imposed by the Einstein metric condition, generalizing known rigidity phenomena outside the strictly pseudoconvex case and providing a foundation for further investigation into canonical metrics and their relation to global domain structure. The methods and conjectural extensions point to substantial potential in further studying the rigidity of solitonic conditions and their algebraic and differential-geometric underpinnings.