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Lagrangian Hofer-Zehnder Capacities and Energy-Capacity Inequalities

Samuel Lisi, Antonio Rieser

TL;DR

The paper defines a modified coisotropic Hofer-Zehnder capacity for Lagrangian submanifolds in monotone symplectic manifolds and proves an energy-capacity inequality $\hat{c}_{HZ}(W,L)\le d(L)$ for displaceable monotone Lagrangians with minimal Maslov number $N_L\ge 2$. The authors employ Floer theory, persistently filtered Floer homology, and local Floer homology to relate displacement energy to a coisotropic capacity, addressing both nondegenerate and degenerate Hamiltonians through a robust r2p persistence framework. A key contribution is showing the necessity of the modified capacity by presenting a counterexample to the unmodified version, and establishing a detailed link between spectral-type invariants and displacement energy. The results extend connections between Lagrangian spectral invariants, Barraud-Cornea/Biran-Cornea perspectives, and energy bounds in a general monotone setting, with potential implications for spectral capacities near Lagrangian submanifolds. Overall, the work provides a Floer-theoretic route to quantitative energy-capacity relations for Lagrangian displaceability.

Abstract

We introduce a new coisotropic Hofer-Zehnder capacity and use it to prove an energy-capacity inequality for displaceable Lagrangians.

Lagrangian Hofer-Zehnder Capacities and Energy-Capacity Inequalities

TL;DR

The paper defines a modified coisotropic Hofer-Zehnder capacity for Lagrangian submanifolds in monotone symplectic manifolds and proves an energy-capacity inequality for displaceable monotone Lagrangians with minimal Maslov number . The authors employ Floer theory, persistently filtered Floer homology, and local Floer homology to relate displacement energy to a coisotropic capacity, addressing both nondegenerate and degenerate Hamiltonians through a robust r2p persistence framework. A key contribution is showing the necessity of the modified capacity by presenting a counterexample to the unmodified version, and establishing a detailed link between spectral-type invariants and displacement energy. The results extend connections between Lagrangian spectral invariants, Barraud-Cornea/Biran-Cornea perspectives, and energy bounds in a general monotone setting, with potential implications for spectral capacities near Lagrangian submanifolds. Overall, the work provides a Floer-theoretic route to quantitative energy-capacity relations for Lagrangian displaceability.

Abstract

We introduce a new coisotropic Hofer-Zehnder capacity and use it to prove an energy-capacity inequality for displaceable Lagrangians.

Paper Structure

This paper contains 14 sections, 13 theorems, 101 equations.

Table of Contents

  1. Introduction and main results
  2. The modified coiostropic Hofer-Zehnder capacity
  3. The modified relative Hofer-Zehnder capacity
  4. Failure of Theorem \ref{['T:displacement']} for the unmodified capacity
  5. Filtered Floer complexes and persistence modules
  6. The Lagrangian pearl complex and Lagrangian Floer homology
  7. Lagrangian pearl complex
  8. Floer homology
  9. Chain maps and consequences
  10. Persistent homology for admissible Hamiltonians
  11. The r2p persistence module in the non-degenerate case
  12. The r2p persistence module in the degenerate case
  13. Local Floer homology
  14. Proof of Theorem \ref{['T:displacement']}

Key Result

Theorem 1.1

Let $(W,\omega)$ be a compact or geometrically bounded symplectic manifold, and suppose that $L \subset (W, \omega)$ is a connected, closed Lagrangian. Suppose, furthermore, that $L$ is monotone with minimal Maslov number at least $2$. Let $d(L)$ denote the displacement energy of the Lagrangian $L$

Theorems & Definitions (25)

  • Theorem 1.1
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • Proposition 2.4
  • Proposition 3.1: Biran_Cornea_pp_2007*Remark 6.6.3
  • Definition 3.2
  • Theorem 3.3
  • Lemma 3.4
  • ...and 15 more