On the spectral diameter of the Grassmannians

TL;DR

This paper investigates the spectral diameter of the Hamiltonian diffeomorphism group on Grassmannians, showing finiteness for prime p and infiniteness for certain even-dimensional cases, revealing geometric complexity differences.

Contribution

It establishes the finiteness of spectral diameter for prime p and the infiniteness for specific even-dimensional Grassmannians, advancing understanding of symplectic geometry.

Findings

Spectral diameter is finite for $ ext{Gr}(2,p)$ with prime p.

Spectral diameter is infinite for $ ext{Gr}(2k,2n)$ with natural numbers k<n.

Highlights geometric differences in symplectic structures of Grassmannians.

Abstract

The diameter of the spectral pseudometric on the universal cover of the Hamiltonian diffeomorphism group of $\mathrm{Gr}(2,p)$ is shown to be finite whenever $p$ is a prime number. On the other hand, it is shown that the diameter is infinite in the case of $\mathrm{Gr}(2k,2n)$ for all natural numbers $k<n$.