Spectral geometry of the group of Hamiltonian symplectomorphisms

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.