A relative Hofer estimate and the asymptotic Hofer-Lipschitz constant

TL;DR

This paper investigates the relationship between Hamiltonian groups of open subsets and entire symplectic manifolds, providing bounds on Hofer norms and applications to Hofer-Lipschitz constants and diameters.

Contribution

It establishes an upper bound for the inclusion of Hamiltonian groups in terms of Hofer norms, with sharp bounds for related symplectic invariants.

Findings

Upper bounds for the inclusion map in Hofer norms

Sharp bounds on the asymptotic Hofer-Lipschitz constant

Sharp or near-sharp bounds on the relative Hofer diameter

Abstract

Let $(M,\omega)$ be a symplectic manifold and $U\subseteq M$ an open subset. We study the natural inclusion of the compactly supported Hamiltonian group of $U$ in the compactly supported Hamiltonian group of $M$. The main result is an upper bound for this map in terms of the Hofer norms for $U$ and $M$. Applications are upper bounds on the asymptotic Hofer-Lipschitz constant and the relative Hofer diameter of $U$. The first bound is often sharp and the second one is often sharp up to a factor of 2.