Observations on the Hofer distance between closed subsets

TL;DR

This paper reveals that the Hofer distance between closed subsets in a symplectic manifold can be characterized by Hamiltonian restrictions, providing insights into energy-capacity inequalities and new vanishing results for specific subsets.

Contribution

It introduces a novel expression for the Hofer distance in terms of Hamiltonian restrictions and extends vanishing results to singular analytic subvarieties.

Findings

Hofer distance can be expressed via Hamiltonian restrictions

Energy-capacity inequalities are better understood through this framework

Vanishing results apply to singular analytic subvarieties

Abstract

We prove the elementary but surprising fact that the Hofer distance between two closed subsets of a symplectic manifold can be expressed in terms of the restrictions of Hamiltonians to one of the subsets; this helps explain certain energy-capacity inequalities that appeared recently in work of Borman-McLean and Humiliere-Leclercq-Seyfaddini. We also build on arXiv:1201.2926 to obtain new vanishing results for the Hofer distance between subsets, applicable for instance to singular analytic subvarieties of Kahler manifolds.