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

TL;DR

This paper uses spectral invariants and Floer theory to analyze Hamiltonian diffeomorphisms on symplectic manifolds, introducing a spectral norm and exploring length minimization properties with geometric analysis techniques.

Contribution

It constructs a spectral norm on the Hamiltonian diffeomorphism group and applies Floer theory to study length minimizing properties of Hamiltonian paths.

Findings

Spectral norm provides new lower bounds for Hamiltonian diffeomorphisms.

Application of Floer theory reveals length minimizing properties.

Development of geometric analysis techniques like adiabatic degeneration.

Abstract

In this paper, we apply spectral invariants, constructed in [Oh5,8], to the study of Hamiltonian diffeomorphisms of closed symplectic manifolds $(M,\omega)$. Using spectral invariants, we first construct an invariant norm called the {\it spectral norm} on the Hamiltonian diffeomorphism group and obtain several lower bounds for the spectral norm in terms of the $\e$-regularity theorem and the symplectic area of certain pseudo-holomorphic curves. We then apply spectral invariants to the study of length minimizing properties of certain Hamiltonian paths {\it among all paths}. In addition to the construction of spectral invariants, these applications rely heavily on the {\it chain level Floer theory} and on some existence theorems with energy bounds of pseudo-holomorphic sections of certain Hamiltonian fibrations with prescribed monodromy. The existence scheme that we develop in this paper in turn relies on some careful geometric analysis involving {\it adiabatic degeneration} and {\it thick-thin decomposition} of the Floer moduli spaces which has an independent interest of its own. We assume that $(M,\omega)$ is {\it strongly semi-positive} throughout, which will be removed in a sequel.