A nonextension result on the spectral metric

TL;DR

This paper demonstrates that the spectral metric cannot be extended to the full symplectomorphism group for certain symplectic manifolds and confirms the bounded isometry conjecture for tori with linear forms.

Contribution

It provides a nonextension result for the spectral metric and verifies the bounded isometry conjecture in specific cases, advancing understanding of symplectic geometry.

Findings

Spectral metric cannot be extended to all symplectomorphisms on certain manifolds.

The bounded isometry conjecture holds for tori with all linear symplectic forms.

The study links Floer theory with symplectic group properties.

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.