Isosystolic inequalities on two-dimensional Finsler tori

TL;DR

This paper surveys known optimal isosystolic inequalities on two-dimensional Finsler tori involving Busemann-Hausdorff and Holmes-Thompson areas, and establishes a new inequality for reversible Finsler tori.

Contribution

It provides a comprehensive survey of existing inequalities and introduces a new optimal inequality for Busemann-Hausdorff area on reversible Finsler tori.

Findings

Survey of all known optimal inequalities involving Finsler areas.

Establishment of a new inequality: Busemann-Hausdorff area ≥ π/4 for reversible 2-tori.

Extension of prior work by Burago and Ivanov.

Abstract

In this article we survey all known optimal isosystolic inequalities on two-dimensional Finsler tori involving the following two central notions of Finsler area: the Busemann-Hausdorff area and the Holmes-Thompson area. We also complete the panorama by establishing the following new optimal isosystolic inequality that is deduced from prior work by Burago and Ivanov: the Busemann-Hausdorff area of a Finsler reversible $2$-torus with unit systole is at least equal to $\pi/4$.