A class of new bi-invariant metrics on the Hamiltonian diffeomorphism groups

TL;DR

This paper introduces a new family of bi-invariant metrics on Hamiltonian diffeomorphism groups, exploring their properties and extending classical inequalities like Hofer's and Sikorav's.

Contribution

It constructs infinitely many such metrics and generalizes key inequalities, advancing the understanding of the geometric structure of Hamiltonian diffeomorphisms.

Findings

Existence of infinitely many bi-invariant metrics

Generalizations of Hofer and Sikorav inequalities

Insights into the geometric properties of Hamiltonian diffeomorphism groups

Abstract

In this paper, we construct infinitely many bi-invariant metrics on the Hamiltonian diffeomorphism group and study their basic properties and corresponding generalizations of the Hofer inequality and Sikorav one.