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

TL;DR

This paper investigates a right invariant metric on the automorphism group of the unit disk, providing an explicit formula and exploring its geometric structure.

Contribution

It derives an explicit formula for the right invariant metric on the automorphism group of the disk and characterizes its Finsler geometric structure.

Findings

Explicit formula for the right invariant metric $d_{H^{eta}}$

Characterization of the Finsler geometric structure

Insights into the automorphism group's geometry

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, $d_{H^{\infty}}(\varphi, \psi)=\sup_{z\in\mathbb{D}}|\varphi(z)-\psi(z)|$ for any $\varphi, \psi\in \rm{Aut}(\mathbb{D})$. We give the explicit formula of the right invariant metric $d_{H^{\infty}}$ and characterize the almost regular Finsler geometric structure of $(\rm{Aut}(\mathbb{D}), d_{H^{\infty}})$.