Spectral invariants, analysis of the Floer moduli space, and geometry of the Hamiltonian diffeomorphism group

Abstract

In this article, we apply spectral invariants constructed in [Oh5] and [6] to the study of Hamiltonian diffeomorphisms of closed symplectic manifolds $(M,\omega )$. Using spectral invariants, we first construct an invariant norm called the spectral norm on the Hamiltonian diffeomorphism group and obtain several lower bounds for the spectral norm in terms of the $\epsilon$-regularity theorem and the symplectic area of certain pseudoholomorphic curves. We then apply spectral invariants to the study of length minimizing properties of certain Hamiltonian paths among all paths. In addition to the construction of spectral invariants, these applications rely heavily on the chain-level Floer theory and on some existence theorems with energy bounds of pseudoholomorphic sections of certain Hamiltonian fibrations with prescribed monodromy. The existence scheme that we develop in this article in turn relies on some careful geometric analysis involving adiabatic degeneration and thick-thin decomposition of the Floer moduli spaces which has an independent interest of its own. We assume that $(M,\omega )$ is strongly semipositive throughout this article