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

In this paper, we study the right invariant metric $d_{H^{\mathrm{\infty }}}$ on the analytic automorphism group $\mathrm{A}\mathrm{u}\mathrm{t}(\mathbb{D})$ of the unit open disk $\mathbb{D}$ induced by maximal modulus, that is, $d_{H^{\mathrm{\infty }}}(\phi ,\psi )={\sup }_{z\in \mathbb{D}}\mathrm{\mid }\phi (z)-\psi (z)\mathrm{\mid }$ for any $\phi ,\psi \in \mathrm{A}\mathrm{u}\mathrm{t}(\mathbb{D})$. We give the explicit formula of the right invariant metric $d_{H^{\mathrm{\infty }}}$ and characterize the almost regular Finsler geometric structure of $(\mathrm{A}\mathrm{u}\mathrm{t}(\mathbb{D}),d_{H^{\mathrm{\infty }}})$.