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

By Yong-Geun Oh. 2004

61 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/0a21c1ab288d2c210cb1eff7f95cc280d0cb3fc6

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Boundary depth in Floer theory and its applications to Hamiltonian dynamics and coisotropic submanifolds (2009, unknown): “09 03 v2 [ m at h. SG ] 1 3 D ec 2”
Spectral invariants for monotone Lagrangians (2015, methodology): “The properties of the Lagrangian spectral invariants listed in Theorem 2 have previously appeared in other contexts, and consequently they are more or less standard, see [Vit92, Oh05a, MVZ12].”
THE SHARP ENERGY-CAPACITY INEQUALITY (2008, methodology): “Thus our argument gives a new proof of the fact that γ vanishes only for the identity symplectomorphism; this was established by a subtle, rather different argument as Theorem A(1) of [20].”