The right invariant metric on the analytic automorphism group of the unit open disk induced by maximal modulus

Abstract: In this paper, we study the right invariant metric $d_{H^{\infty}}$ on the analytic automorphism group $\rm{Aut}(\mathbb{D})$ of the unit open disk $\mathbb{D}$ induced by maximal modulus, that is, $d_{H^{\infty}}(\varphi, ψ)=\sup_{z\in\mathbb{D}}|\varphi(z)-ψ(z)|$ for any $\varphi, ψ\in \rm{Aut}(\mathbb{D})$. We give the explicit formula of the right invariant metric $d_{H^{\infty}}$ and characterize the almost regular Finsler geometric structure of $(\rm{Aut}(\mathbb{D}), d_{H^{\infty}})$.

• The paper derives an explicit closed-form distance formula for the maximal modulus-induced right invariant metric on Aut(D), detailing its derivation and geometric implications.
• It employs a detailed Finsler framework to expose a non-Riemannian, almost regular structure with properties similar to Randers metrics on the Lie group.
• The findings reveal non-geodesicity in the metric space of Aut(D), suggesting significant avenues for further exploration in curvature analysis and complex function theory.

Right Invariant Metric Structures on the Analytic Automorphism Group of the Unit Disk

Introduction and Mathematical Setting

This paper addresses the explicit construction and analysis of a right invariant metric $d_{H^\infty}$ on the analytic automorphism group $\mathrm{Aut}(\mathbb{D})$ of the unit open disk $\mathbb{D}$, where $d_{H^\infty}$ is induced by the maximal modulus norm. Explicitly, for $\varphi, \psi \in \mathrm{Aut}(\mathbb{D})$, $$d_{H^\infty}(\varphi, \psi) = \sup_{z \in \mathbb{D}} |\varphi(z) - \psi(z)|$$.

The work situates $\mathrm{Aut}(\mathbb{D})$ as a finite-dimensional Lie group isomorphic to $\mathrm{PSL}(2, \mathbb{R})$ and investigates how this functional-analytic metric endows the group with a non-Riemannian Finsler geometric structure, with a particular focus on the almost regular case, where regularity fails on a proper subbundle. The connection to Randers metrics, a subclass of Finsler metrics, is also established.

Main Results: Explicit Distance Formula and Structure

The principal achievement of this paper is the derivation of a closed-formula for $d_{H^\infty}$ and an analysis of the resulting geometric and topological structures. The automorphisms can be parametrized as $$f_{\xi, u}(z) = e^{i\xi} \frac{z - u}{1 - \overline{u} z}$$ for $\mathrm{Aut}(\mathbb{D})$0, $\mathrm{Aut}(\mathbb{D})$1. The authors prove:

Explicit Formula for the Distance

Given $\mathrm{Aut}(\mathbb{D})$2, set $\mathrm{Aut}(\mathbb{D})$3 and $\mathrm{Aut}(\mathbb{D})$4.

• If $\mathrm{Aut}(\mathbb{D})$5, then $\mathrm{Aut}(\mathbb{D})$6.
• If $\mathrm{Aut}(\mathbb{D})$7, then

$\mathrm{Aut}(\mathbb{D})$8

This formula is derived through a careful analysis of the supremal modulus difference on orbits in the disk and their geometric relation to Möbius transformations.

Finsler Geometry and Regularity Analysis

The metric $\mathrm{Aut}(\mathbb{D})$9 induces a non-Riemannian, reversible, almost regular Finsler structure $\mathbb{D}$0 on $\mathbb{D}$1. The tangent space at $\mathbb{D}$2 is coordinatized by pairs $\mathbb{D}$3, corresponding to infinitesimal angular and translational directions. The norm is given by:

$\mathbb{D}$4

• Reversibility follows from the symmetry of the formula in $\mathbb{D}$5.
• Non-Riemannianity is certified by the non-quadratic dependence of $\mathbb{D}$6 on tangent directions.
• Almost regularity results from smoothness except along a subbundle defined by $\mathbb{D}$7.
• Randers structure: On the submanifold $\mathbb{D}$8, the induced Finsler metric $\mathbb{D}$9 is of the classical almost regular Randers type.

These findings extend known constructions of left/right-invariant Finsler metrics on Lie groups, particularly in settings where classical Riemannian geometry is insufficient to capture the metric subtleties.

Geodesic and Topological Properties

A fundamental implication is that $d_{H^\infty}$0 is not a geodesic space; for certain pairs of automorphisms, there is no geodesic (in the metric sense) connecting them whose length realizes the distance. This irregularity results from the maximal modulus structure and is geometrically manifested in the existence of open sets where $d_{H^\infty}$1 is identically constant.

The submanifold $d_{H^\infty}$2, isometrically corresponding to the disk equipped with the induced metric, exhibits analogous phenomena. Intersections with certain ellipsoidal loci demarcate constant-distance regions, again precluding geodesic completeness.

Implications and Future Directions

This work situates the $d_{H^\infty}$3-right invariant metric as a natural yet highly non-Euclidean object for the complex geometric group $d_{H^\infty}$4, with several implications:

• In complex analysis and operator theory, the structure underscores the sharp distinctions between various metrics (e.g., Carathéodory, Kobayashi, and maximal modulus).
• In group geometry, it supplies a concrete almost regular Finsler structure on a Lie group, enriching the classification and geometric taxonomy of invariant Finsler metrics.
• The lack of geodesicity indicates that the $d_{H^\infty}$5 topology is topologically and geometrically singular, suggesting that further exploration of geodesically convex or geodetic completions could be fruitful.
• The explicit connection to almost regular Randers metrics, together with their reversibility and non-Riemannianity, positions this construction as a testbed for the study of Finslerian curvature and geometric flow behaviors that sharply depart from the Riemannian context.

Future directions encompass a detailed curvature analysis of $d_{H^\infty}$6, its applications in function theory (e.g., in the context of function space isometries), and the study of similar structures on other bounded symmetric domains or infinite-dimensional analogs, such as automorphism groups of Hardy/Bergman spaces.

Conclusion

The paper rigorously characterizes the right invariant maximal modulus metric on the analytic automorphism group of the disk, detailing its explicit formula, revealing its non-Riemannian and almost regular Finslerian structure, and establishing topological properties such as non-geodesicity. The work contributes novel explicit metric and geometric perspectives on function group automorphisms and extends the toolkit for analyzing their Finsler and metric properties within complex analysis and differential geometry (2604.19583).