Three isometrically equivalent models of the Finsler-Poincaré disk

We present the isometry between the 2-dimensional Funk model and the Finsler-Poincaré disk. Then, we introduce the Finslerian Poincaré upper half plane model, which turns out to be also isometrically equivalent to the previous models. As application, we state the gapless character of the first eigenvalue for the aforementioned three spaces.