Quasimorphisms on surfaces and continuity in the Hofer norm

TL;DR

This paper investigates the compatibility of various quasimorphisms on Hamiltonian groups of surfaces with the Hofer norm, revealing most are not continuous except for specific known cases like the Calabi quasimorphism.

Contribution

It demonstrates that many known quasimorphisms on surface Hamiltonian groups are not continuous with respect to the Hofer norm, except for the Calabi quasimorphism on spheres and genus-zero surfaces.

Findings

Most quasimorphisms are not Lipschitz with the Hofer norm

The Calabi quasimorphism on spheres remains continuous and Lipschitz

Continuity is limited to specific known quasimorphisms

Abstract

There is a number of known constructions of quasimorphisms on Hamiltonian groups. We show that on surfaces many of these quasimorphisms are not compatible with the Hofer norm in a sense they are not continuous and not Lipschitz. The only exception known to the author is the Calabi quasimorphism on a sphere and the induced quasimorphisms on genus-zero surfaces.