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Symplectic capacities of disc cotangent bundles of flat tori

Gabriele Benedetti, Johanna Bimmermann, Kai Zehmisch

TL;DR

The paper computes normalized symplectic capacities for the unit disc cotangent bundles of flat, reversible Finsler tori by proving they all equal $2\,\mathrm{sys}(F)$ under broad conditions. The approach combines a SL$(n,\mathbb{Z})$-type reduction to a standard form with a two-part bound: an upper bound obtained via symplectic embeddings into a cylinder, and a lower bound derived from a $(s/2)K\times K^*$ inclusion and a case analysis (domination by a flat metric or a Hofer–Zehnder capacity condition). The key result extends Jiang’s $\ell^1$-torus findings to the flat torus setting and shows that for Riemannian-induced norms (or $\ell^p$ with $p\le 2$), all normalized capacities coincide with $2\,\mathrm{sys}(F)$. The work highlights a concrete instance where the strong Viterbo-type questions about capacity coincidences can be resolved and suggests potential extensions to twisted disc cotangent bundles with magnetic fields.

Abstract

We show that on the unit disc cotangent bundle of flat Riemannian tori, all normalized capacities coincide with twice the systole. The same result holds for flat, reversible Finsler tori and normalized capacities that are greater than or equal to the Hofer-Zehnder capacity.

Symplectic capacities of disc cotangent bundles of flat tori

TL;DR

The paper computes normalized symplectic capacities for the unit disc cotangent bundles of flat, reversible Finsler tori by proving they all equal under broad conditions. The approach combines a SL-type reduction to a standard form with a two-part bound: an upper bound obtained via symplectic embeddings into a cylinder, and a lower bound derived from a inclusion and a case analysis (domination by a flat metric or a Hofer–Zehnder capacity condition). The key result extends Jiang’s -torus findings to the flat torus setting and shows that for Riemannian-induced norms (or with ), all normalized capacities coincide with . The work highlights a concrete instance where the strong Viterbo-type questions about capacity coincidences can be resolved and suggests potential extensions to twisted disc cotangent bundles with magnetic fields.

Abstract

We show that on the unit disc cotangent bundle of flat Riemannian tori, all normalized capacities coincide with twice the systole. The same result holds for flat, reversible Finsler tori and normalized capacities that are greater than or equal to the Hofer-Zehnder capacity.
Paper Structure (4 sections, 1 theorem, 18 equations)

This paper contains 4 sections, 1 theorem, 18 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['thm2']}
  3. Upper bound
  4. Lower bound

Key Result

Theorem 1.1

If $F$ is a flat, reversible Finsler norm on $\mathbb{T}^n$ and $c$ is a (normalized) symplectic capacity, then If, in addition, then

Theorems & Definitions (5)

  • Theorem 1.1
  • Remark 1.2
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3