Pinwheels as Lagrangian barriers

TL;DR

This paper investigates pinwheels, special Lagrangian complexes in CP^2, demonstrating they act as barriers by reducing the Gromov width of their complements, with results linked to the Lagrange spectrum.

Contribution

It computes the Gromov width of pinwheel complements in CP^2 and establishes their role as Lagrangian barriers, connecting symplectic geometry with number theory.

Findings

Gromov width of pinwheel complements is smaller than that of CP^2.

Pinwheels serve as Lagrangian barriers in symplectic geometry.

The Gromov widths' accumulation points relate to the Lagrange spectrum below 3.

Abstract

The complex projective plane CP^2 contains certain Lagrangian CW-complexes called pinwheels, which have interesting rigidity properties related to solutions of the Markov equation. We compute the Gromov width of the complement of pinwheels and show that it is strictly smaller than the Gromov width of CP^2, meaning that pinwheels are Lagrangian barriers in the sense of Biran. The accumulation points of the set of these Gromov widths are in a simple bijection with the Lagrange spectrum below 3.