Isosystolic inequalities on two-dimensional Finsler tori

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 π=4.