The codisc radius capacity

TL;DR

This paper introduces a new capacity to measure Weinstein tubular neighborhoods of Lagrangian submanifolds, providing bounds on the codisc radius and generalizing Gromov's packing inequality, with applications to symplectic embeddings.

Contribution

It defines a novel capacity related to Weinstein neighborhoods, extends Gromov's packing inequality, and establishes finiteness results for the spherical Gromov radius of monotone Lagrangian tori.

Findings

Bounds on codisc radius via Viterbo's inequality

Generalized Gromov packing inequality for symplectic embeddings

Finiteness of spherical Gromov radius for monotone Lagrangian tori

Abstract

We define a capacity which measures the size of Weinstein tubular neighbourhoods of Lagrangian submanifolds. In symplectic vector spaces this leads to bounds on the codisc radius for any closed Lagrangian submanifold in terms of Viterbo's isoperimetric inequality. Moreover, we prove a generalization of Gromov's packing inequality concerning symplectic embeddings of the boundaries of two balls of equal radius into the open unit ball. If the interior components of the image spheres are disjoint, then the radii are less than the square root of one half. Furthermore, we introduce the spherical variant of the relative Gromov radius and prove finiteness for monotone Lagrangian tori in symplectic vector spaces.