Schwarz Lemma for Bounded Domains

• Schwarz Lemma for bounded domains is a rigidity result that controls the distortion of the Bergman metric under holomorphic maps using analytic and geometric techniques.
• It derives explicit constants, generalizing the classical Schwarz–Pick lemma through sharp inequalities on canonical kernels.
• Probabilistic and information‐geometric methods underpin the approach, offering a robust framework for metric comparison with minimal assumptions.

A Schwarz Lemma for bounded domains equipped with Bergman metrics is a rigidity result controlling the distortion of the Bergman metric under holomorphic mappings. It generalizes the classical Schwarz–Pick lemma from the unit disk to several variables and replaces curvature or pseudoconvexity hypotheses with analytic bounds on canonical kernels. Recent work has developed sharp forms and information-geometric methods, leading to metric comparison and rigidity theorems with minimal assumptions.

1. Bergman Kernel and Metric on Bounded Domains

Let $\Omega \subset \mathbb{C}^n$ be a bounded domain. The Bergman space $A^2(\Omega)$ consists of holomorphic functions square-integrable with respect to Lebesgue measure. The orthonormal basis $\{\phi_j\}$ yields the Bergman kernel $B(z,\xi) = \sum_j \phi_j(z)\overline{\phi_j(\xi)}$ with $B(z,z) > 0$. The Kähler metric derived from the kernel,

$$g_{i\bar j}(z) = \frac{\partial^2}{\partial z_i \partial\bar z_j} \log B(z,z),$$

is the Bergman metric, real-analytic and invariant under biholomorphisms. In this context, bounded domains are not required to be pseudoconvex or have smooth boundary to admit these structures, but $B(z,z) > 0$ everywhere is essential for nondegeneracy (Seo et al., 30 Dec 2025).

2. The Schwarz Lemma for Bergman Metrics: Main Statement

The new Schwarz Lemma states:

Theorem A (Seo et al., 30 Dec 2025): Let $(\Omega_1,g_{B_1}) \subset \mathbb{C}^n$ and $(\Omega_2,g_{B_2}) \subset \mathbb{C}^m$ be bounded domains with Bergman metrics. Suppose there exists $C > 0$ such that

$$\sup_{w,\,\zeta \in \Omega_2} |\partial_w \log P_2(w,\zeta)|^2_{g_{B_2}} \leq C,$$

where $P_2(w,\zeta) = |B_2(w,\zeta)|^2/B_2(w,w)$ is the Poisson–Bergman kernel. Then for any holomorphic map $f \colon \Omega_1 \to \Omega_2$,

$f^* g_{B_2} \leq C\,g_{B_1},$

holds as Hermitian forms.

This is analogous to the classical Schwarz–Pick lemma and provides a sharp, explicit bound for the distortion of the Bergman metric under holomorphic maps in terms of a supremum over gradients of the log-Poisson kernel of the target domain.

3. Probabilistic and Information-Geometric Techniques

The proof uses probabilistic methods and the geometric structure of statistical models. For each $w \in \Omega_2$, the function $P_2(w,\cdot)$ is a probability density. Considering tangent vectors $X$ at $z \in \Omega_1$, define random variables

$$Z(\xi) = \partial_X \log P_1(z,\xi),\quad W(\xi) = \partial_X \log P_2(f(z),f(\xi)).$$

Then

$$g_{B_1}(X,X) = \operatorname{Var}[Z],\quad f^* g_{B_2}(X,X) = \operatorname{Cov}[Z,W].$$

Using the Cauchy–Schwarz inequality for covariance,

$$|f^* g_{B_2}(X,X)|^2 \leq g_{B_1}(X,X) \cdot \operatorname{Var}[W].$$

The uniform bound on $|\partial_w \log P_2|^2_{g_{B_2}}$ controls $\operatorname{Var}[W]$, leading to the key inequality above (Seo et al., 30 Dec 2025). This approach exposes a deep link to the Fisher information metric in probability theory, and more generally, Information geometry offers alternative rigidity proofs in this context (Cho et al., 2023, Yum, 2024).

4. Explicit Constants and Examples

On the unit ball $B^n \subset \mathbb{C}^n$, the Bergman kernel has the explicit form

$$B(z,\xi) = \frac{n!}{\pi^n}(1-\langle z,\xi \rangle)^{-(n+1)},$$

so

$|\partial_w \log P(w,\zeta)|^2_{g_B} \equiv n+1,$

yielding the sharp Schwarz Lemma constant $C = n+1$. For polydisks $\Delta^n$, $C = 2n$. These values match and generalize well-known metric comparison results.

Homogeneous bounded domains also admit such uniform bounds by passing to their Siegel realizations and exploiting metric invariance (Seo et al., 30 Dec 2025).

5. Rigidity, Equality Cases, and Biholomorphic Maps

In the equality case (i.e., when $f^* g_{B_2} = C g_{B_1}$ everywhere), information-geometric characterization applies: a holomorphic local isometry of Bergman metrics (up to a scale) between bounded domains, which is proper, must be a biholomorphism (Yum, 2024). The factorization theorem from Information geometry shows that the pushforward of measures via $f$ preserves Fisher information if and only if $f$ is a sufficient statistic, which for holomorphic maps corresponds to being a biholomorphism (Cho et al., 2023).

6. Extensions, Comparison with Metric Schwarz Lemmas, and Open Questions

Unlike classic metric Schwarz lemmas (Yau, Royden), which require strong metric curvature bounds (Ricci lower, bisectional or holomorphic sectional upper), the analytic Schwarz Lemma for Bergman metrics needs only a supremum over Poisson–Bergman kernel gradients; no curvature assumption or completeness is imposed on the source domain.

Open problems involve:

• Geometric interpretation of the kernel-gradient bound: Is the boundedness of $|\partial_w \log P_2|^2$ strictly weaker than any negative curvature or Kähler hyperbolicity, or is it equivalent?
• Classification of extremal cases for equality and $\lambda$-isometries.
• Potential extension to other canonical metrics (Carathéodory, Sibony's P-metric, etc.) (Seo et al., 30 Dec 2025).
• Refinements replacing supremum conditions by $L^p$ integrability or other analytic bounds.

7. Impact on Complex Geometry and Further Directions

The Schwarz Lemma for Bergman metrics introduces a new methodology for metric comparison and rigidity utilizing canonical kernels, probability theory, and information geometry. It generalizes classical results, is sharp on canonical examples, and is robust under minimal regularity and boundary conditions. Its connection to Fisher information, sufficiency, and the statistical structure of holomorphic kernels points toward further applications in several complex variables, geometric quantization, and even statistical learning theory in geometric settings (Cho et al., 2023).

Key references:

• "On the Schwarz Lemma for Bergman metrics of bounded domains" (Seo et al., 30 Dec 2025) (novel probabilistic approach, explicit bound statements)
• "Bergman local isometries are biholomorphisms" (Yum, 2024) (information geometry and rigidity)
• "Statistical Bergman geometry" (Cho et al., 2023) (Bergman/Fisher equivalence, probabilistic curvature formulas)