Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Isosystolic inequalities on two-dimensional Finsler tori

Florent Balacheff, Teo Gil Moreno de Mora

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 $π/4$.

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 -torus with unit systole is at least equal to .
Paper Structure (23 sections, 18 theorems, 70 equations, 3 figures, 1 table)

This paper contains 23 sections, 18 theorems, 70 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Finsler metrics
  3. Systole and Finsler areas
  4. Optimal isosystolic inequalities
  5. Organization of the paper.
  6. Isosystolic inequalities for flat and non-flat Riemannian metrics
  7. The flat Riemannian case: Hermite constant in dimension $2$.
  8. The Riemannian case: Loewner's isosystolic inequality
  9. Isosystolic inequalities for flat Finsler metrics
  10. Busemann-Hausdorff area in the flat reversible case
  11. Busemann-Hausdorff area in the flat non-reversible case: systolic freedom
  12. Holmes-Thompson area in the flat reversible case
  13. Holmes-Thompson area in the flat non-reversible case
  14. Asymptotic geometry in the universal cover of a Finsler $2$-torus
  15. Stable norm
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let $F$ be a Finsler reversible metric on $\mathbb{T}^2$. Then the following optimal inequality holds true: Equality holds for the flat metric corresponding to the supremum norm $\|\cdot\|_\infty$.

Figures (3)

  • Figure 1: The convex body $K_\varepsilon$.
  • Figure 2: Set of integer lines not parallel to the axes
  • Figure 3: Deformation of the convex polygon.

Theorems & Definitions (37)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.1
  • Theorem 2.1: Folklore
  • proof
  • Theorem 2.2: Hermite constant in dimension $2$, systolic formulation
  • Theorem 2.3: Loewner's isosystolic inequality, 1949
  • proof
  • Theorem 3.1: Minkowski's first theorem, 1896
  • proof
  • ...and 27 more