Relative Hofer Geometry and the Asymptotic Hofer-Lipschitz Constant

Abstract

Let $(M,\omega)$ be a symplectic manifold and $U\subseteq M$ an open subset. I study the natural inclusion of the group of Hamiltonian diffeomorphisms of $U$ into the group of Hamiltonian diffeomorphisms 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 relative Hofer diameter of $U$ and the asymptotic Hofer-Lipschitz constant, which are often sharp up to constant factors. As another consequence, the relative Hofer diameter of certain symplectic submanifolds vanishes.