On the Lagrangian Hofer geometry of Clifford tori

TL;DR

This paper demonstrates that the space of Hamiltonian isotopic Clifford tori in certain symplectic manifolds contains an infinite diameter quasi-isometric copy of the real line, revealing complex geometric structure.

Contribution

It establishes the infinite diameter and quasi-isometric embedding of the real line in the Lagrangian Hofer metric space of Clifford tori.

Findings

The space has infinite diameter.

Contains a quasi-isometric copy of the real line.

Reveals complex geometric structure of Lagrangian space.

Abstract

We show that the space of Lagrangians which are Hamiltonian isotopic to the Clifford torus in a complex projective space or in the four-dimensional quadric, taken with Chekanov's Lagrangian Hofer metric, contains a quasi-isometric copy of the real line, and in particular has infinite diameter.