A nonextension result on the spectral metric

2008/05/07 by Zhigang Han, Han, Zhigang
Mathematics · #53D35 #53D40 #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Symplectic Geometry (math.SG)

paper · pdf · doi:10.48550/arxiv.0805.1045

openalex publication_date 2008/05/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The spectral metric, defined by Schwarz and Oh using Floer-theoretical method, is a bi-invariant metric on the Hamiltonian diffeomorphism group. We show in this note that for certain symplectic manifolds, this metric can not be extended to a bi-invariant metric on the full group of symplectomorphisms. We also study the bounded isometry conjecture of Lalonde and Polterovich in the context of the spectral metric. In particular, we show that the conjecture holds for the torus with all linear symplectic forms.

Citations

• Symplectic manifolds and their lagrangian submanifolds
• Monodromy in Hamiltonian Floer theory
• The bounded isometry conjecture for the Kodaira-Thurston manifold and 4-Torus

Related

• Bi-invariant metrics on the group of symplectomorphisms
• A class of new bi-invariant metrics on the Hamiltonian diffeomorphism groups
• Spectral invariants over the integers
• A Hofer-like metric on the group of symplectic diffeomorphisms
• Bounded symplectic diffeomorphisms and split flux groups