Three isometrically equivalent models of the Finsler-Poincar\'e disk

TL;DR

This paper establishes isometric equivalences among three Finsler models of the Poincaré disk, introduces a new Finslerian upper half-plane model, and discusses spectral properties of these spaces.

Contribution

It demonstrates the isometry between three Finsler models of the Poincaré disk and introduces a new Finslerian upper half-plane model.

Findings

The three models are isometrically equivalent.

The Finslerian upper half-plane model is introduced and shown to be equivalent.

The first eigenvalue is gapless in all three models.

Abstract

We present the isometry between the 2-dimensional Funk model and the Finsler-Poincar\'e disk. Then, we introduce the Finslerian Poincar\'e 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.