Spectral geometry of the group of Hamiltonian symplectomorphisms

TL;DR

This paper introduces a spectral length functional for Hamiltonian symplectomorphisms, explores its properties, and provides evidence of its variational completeness, especially on the sphere, linking it to spectral invariants and norms.

Contribution

It defines a new spectral length functional on Hamiltonian paths, investigates its properties, and demonstrates its potential variational completeness, connecting it to spectral invariants and norms.

Findings

Spectral length functional is smooth and potentially variationally complete.

All Lalonde-McDuff Hamiltonian symplectomorphisms on S^2 are connected to identity by minimizing paths.

The associated norm is non-degenerate and bounded below by the spectral norm.

Abstract

We introduce here a natural functional associated to any $b \in QH_* (M, \omega)$: \emph{spectral length functional}, on the space of "generalized paths" in $ \text {Ham}(M, \omega)$, closely related to both the Hofer length functional and spectral invariants and establish some of its properties. This functional is smooth on its domain of definition, and moreover the nature of extremals of this functional suggests that it may be variationally complete, in the sense that any suitably generic element of $ \widetilde{\text {Ham}}(M, \omega)$ is connected to $id$ by a generalized path minimizing spectral length. Rather strong evidence is given for this when $M=S ^{2}$, where we show that all the Lalonde-McDuff Hamiltonian symplectomorphisms are joined to id by such a path. We also prove that the associated norm on $ {\text {Ham}}(M, \omega)$ is non-degenerate and bounded from below by the the spectral norm. If the spectral length functional is variationally complete the associated norm reduces to the spectral norm.