Pinwheels as Lagrangian barriers

Abstract

The complex projective plane [Formula: see text] contains certain Lagrangian CW-complexes called pinwheels, which have interesting rigidity properties related to solutions of the Markov equation, see for example [J. Evans and I. Smith, Markov numbers and Lagrangian cell complexes in the complex projective plane, Geom. Topol. 22 (2018) 1143–1180]. We compute the Gromov width of the complement of pinwheels and show that it is strictly smaller than the Gromov width of [Formula: see text], meaning that pinwheels are Lagrangian barriers in the sense of [P. Biran, Lagrangian barriers and symplectic embeddings, Geom. Funct. Anal. 11 (2001) 407–464]. The accumulation points of the set of these Gromov widths are in a simple bijection with the Lagrange spectrum below [Formula: see text].