Morse theory for the Hofer length functional

TL;DR

This paper establishes a novel connection between Morse theory for the Hofer length functional and Gromov-Witten/Floer theory on monotone symplectic manifolds, revealing topological restrictions and non-existence criteria for certain geodesics.

Contribution

It introduces a new approach linking Hofer geometry with Gromov-Witten/Floer theory, utilizing automatic transversality specific to monotone symplectic manifolds.

Findings

Restrictions on the topology of Hamiltonian symplectomorphism groups

Criteria for the non-existence of certain higher index geodesics

Development of a transversality technique leveraging Hofer geometry

Abstract

Following \cite{citeSavelyevVirtualMorsetheoryon$Omega$Ham$(Momega)$.}, we develop here a connection between Morse theory for the (positive) Hofer length functional $L: \Omega \text {Ham}(M, \omega) \to \mathbb{R}$, with Gromov-Witten/Floer theory, for monotone symplectic manifolds $ (M, \omega) $. This gives some immediate restrictions on the topology of the group of Hamiltonian symplectomorphisms (possibly relative to the Hofer length functional), and a criterion for non-existence of certain higher index geodesics for the Hofer length functional. The argument is based on a certain automatic transversality phenomenon which uses Hofer geometry to conclude transversality and may be useful in other contexts. Strangely the monotone assumption seems essential for this argument, as abstract perturbations necessary for the virtual moduli cycle, decouple us from underlying Hofer geometry, causing automatic transversality to break.