Shortening the Hofer length of Hamiltonian circle actions

TL;DR

This paper investigates how Hamiltonian circle actions on symplectic manifolds can be deformed to shorter loops in the Hofer metric, providing bounds and conditions for such shortenings.

Contribution

It introduces new bounds and conditions for shortening Hamiltonian circle actions in the Hofer metric, extending understanding of their geometric properties.

Findings

Shortening of Hamiltonian circle actions is possible under specific isotropy weight conditions.

Lower bounds are established for the amount of shortening and the number of independent shortening directions.

Deformations are possible when the minimum or maximum is attained along a submanifold with large isotropy weights and un-twisted normal bundle.

Abstract

A Hamiltonian circle action on a compact symplectic manifold is known to be a closed geodesic with respect to the Hofer metric on the group of Hamiltonian diffeomorphisms. If the momentum map attains its minimum or maximum at an isolated fixed point with isotropy weights not all equal to plus or minus one, then this closed geodesic can be deformed into a loop of shorter Hofer length. In this paper we give a lower bound for the possible amount of shortening, and we give a lower bound for the index ("number of independent shortening directions"). If the minimum or maximum is attained along a submanifold B, then we deform the circle action into a loop of shorter Hofer length whenever the isotropy weights have sufficiently large absolute values and the normal bundle of B is sufficiently un-twisted.