[Paper Review] Schwarz lemma and Kobayashi metrics for holomorphic and pluriharmonic functions

Miodrag Mateljević|arXiv (Cornell University)|Apr 21, 2017

Holomorphic and Operator Theory14 references3 citations

TL;DR

This paper generalizes the classical Schwarz lemma to holomorphic and pluriharmonic functions in several complex variables using the Kobayashi metric and hyperbolic geometry. It establishes a geometric form of the Kobayashi-Schwarz lemma that unifies and extends recent results on hyperbolic domains such as the unit ball, polydisc, punctured disk, and strip, providing sharp estimates for derivatives via invariant metrics and invariant Laplacians.

ABSTRACT

The Schwarz lemma as one of the most influential results in complex analysis and it has a great impact to the development of several research fields, such as geometric function theory, hyperbolic geometry, complex dynamical systems, and theory of quasi-conformal mappings. In this note we mainly consider various version of Schwarz lemma and its relatives related to holomorphic functions including several variables.

Motivation & Objective

• To extend classical Schwarz lemma results to several complex variables using Kobayashi geometry.
• To unify recent results by Kalaj-Vuorinen, Chen, Dyakanov, and others under a common framework.
• To establish a geometric form of the Kobayashi-Schwarz lemma applicable to domains like the unit ball, polydisc, punctured disk, and strip.
• To analyze the behavior of holomorphic and pluriharmonic functions via invariant metrics and the invariant Laplacian.
• To derive sharp derivative estimates using hyperbolic and pseudo-hyperbolic distances in complex domains.

Proposed method

• Uses the Ahlfors-Schwarz lemma as a foundational tool to derive derivative bounds for holomorphic maps.
• Introduces the geometric form of the Kobayashi-Schwarz lemma (Theorem 4) to generalize inequalities to several variables.
• Applies the invariant Laplacian operator $\tilde{\triangle}$ to study $\mathcal{M}$-harmonic and pluriharmonic functions on the unit ball.
• Employs automorphism-invariant metrics and the hyperbolic density $\mathrm{Hyp}_G(z)$ to compare tangent vector norms under holomorphic maps.
• Derives estimates via the transformation $F = \varphi_{w_1} \circ f \circ \varphi_{z_1}$, reducing general cases to zero-derivative problems.
• Uses the invariant gradient $\tilde{D}f(z)$ and relates it to the standard gradient $Df(z)$ through the factor $s_z = \frac{1 - |z|^2}{2}$.

Experimental results

Research questions

• RQ1 How can the classical Schwarz lemma be generalized to holomorphic maps between hyperbolic domains in $\mathbb{C}^n$?
• RQ2 What is the role of the Kobayashi metric in deriving sharp derivative estimates for holomorphic and pluriharmonic functions?
• RQ3 How do invariant metrics and the invariant Laplacian $\tilde{\triangle}$ help unify recent results on the punctured disk and strip?
• RQ4 Can the Ahlfors-Schwarz lemma be adapted to yield optimal bounds for maps into domains with non-constant curvature?
• RQ5 What are the precise relationships between the invariant gradient $\tilde{D}f(z)$, the standard gradient $Df(z)$, and the hyperbolic metric?

Key findings

• The geometric form of the Kobayashi-Schwarz lemma (Theorem 4) establishes that for $f: \mathbb{B}_n \to G$, $\mathrm{Hyp}_G(fz) \cdot |f'(z)| \leq \frac{1}{s_z^2}$, where $s_z = \frac{1 - |z|^2}{2}$.
• For $f \in \mathcal{O}(\mathbb{B}_n, \mathbb{U}^\prime)$ with $\mathbb{U}^\prime$ the punctured disk, the inequality $ (1 - |a|^2) |f'(a)| \leq 2|b| \ln \frac{1}{|b|} $ holds, recovering a result of Dyakanov.
• The invariant gradient satisfies $ s_z^2 |Df(z)| \leq |\tilde{D}f(z)| \leq s_z |Df(z)| $, linking the standard and invariant derivatives.
• For $f: \mathbb{B}_n \to G$, the inequality $ \mathrm{Hyp}_G(fz) \cdot |u_*|_e \leq M_{\mathbb{B}_2}(a,u) \cdot |u|_e $ holds for tangent vectors $u$, generalizing the pointwise derivative bound.
• The result extends to pluriharmonic functions $u: \mathbb{B}_n \to (a,b)^m$ via the invariant Laplacian and hyperbolic geometry.
• The framework unifies results from Kalaj-Vuorinen, Chen, Dyakanov, and Melentijević by embedding them in a single invariant metric approach.