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 $\mathrm{Aut}(\mathbb{D})$ of the unit open disk $\mathbb{D}$ induced by maximal modulus, that is, $d_{H^{\infty }}(\phi ,\psi )={\sup }_{z\in \mathbb{D}}|\phi (z)-\psi (z)|$ for any $\phi ,\psi \in \mathrm{Aut}(\mathbb{D})$. We give the explicit formula of the right invariant metric $d_{H^{\infty }}$ and characterize the almost regular Finsler geometric structure of $(\mathrm{Aut}(\mathbb{D}),{\mathrm{d}}_{{\mathrm{H}}^{\infty }})$.

11 pages, 2 figures