Paper · Oct 7, 2025

On the Bergman metric of Cartan-Hartogs domains

Andrea Loi, Roberto Mossa, Fabio ZUddas

math.CVarXiv:2510.06405
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Abstract

We study the Bergman metric and introduce the Bergman dual on Cartan-Hartogs (CH) domains. For a bounded domain D in C^n with Bergman kernel K_D, we define the Bergman dual of (D, g_D) as (D*, g_D*), where D* is the maximal domain on which the modified kernel K_D*(z, zbar) = K_D(z, -zbar) is positive, and g_D* is the Kahler metric obtained from K_D*. For a Cartan-Hartogs domain M_Omega, mu we prove the equivalence of: (i) M_Omega, mu is biholomorphic to the unit ball; (ii) its Bergman metric is a Kahler-Ricci soliton; (iii) after rescaling by a constant factor, the Bergman dual is finitely projectively induced. Conditions (i) and (ii) are Bergman-metric analogues of classical rigidity for Kahler-Einstein metrics (related to Yau's problem and Cheng's conjecture) and to recent rigidity for Kahler-Ricci solitons. Condition (iii) emphasizes the duality viewpoint, inspired by bounded symmetric domains and their compact duals. We also compare our results with other canonical metrics on CH domains, namely g_Omega, mu and hat g_Omega, mu, and discuss open problems about the maximal domain on which the Bergman dual is defined.

Comments: The paper consists of 19 pages

Ricci solitonsKähler–Ricci solitonsEinstein metricsKähler geometryStability & uniquenessKähler–Einstein & K-stability

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Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Ricci solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988) Kähler–Ricci solitons Kähler metrics whose Ricci form differs from the metric by a Lie derivative along a holomorphic vector field. Start with: Uniqueness of Kähler-Ricci solitons (Tian et al., 2000) Einstein metrics Metrics whose Ricci curvature is a constant multiple of the metric. They are the fixed points of normalised Ricci flow, and the most natural metrics… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) Kähler geometry Complex manifolds with compatible Riemannian metrics. Here Ricci curvature is the complex Hessian of a log-determinant, so Ricci flow becomes a… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978)

In the guidebook

  1. 11B.1
    Ricci SolitonsBook 11B · Solitons, Compactness and Singularities

    Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.

  2. 12B.2
    The Asymptotic Soliton and Three-Dimensional ShrinkersBook 12B · κ-Solutions and Surgery

    Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.