# Maximal ball packings of symplectic-toric manifolds

## TL;DR

This paper studies symplectic ball packings in symplectic-toric manifolds, revealing a rich variety of maximal packing densities and showing the set of packings forms a convex polytope.

## Contribution

It demonstrates the existence of uncountably many inequivalent manifolds with specified packing densities and analyzes how these densities vary within families of equivariantly diffeomorphic manifolds.

## Key findings

- The set of toric symplectic ball packings forms a convex polytope.
- For each density r in (0,1), uncountably many manifolds have maximal packings of density r.
- Density of maximal packings can vary continuously within certain families of manifolds.

## Abstract

Let M be a symplectic-toric manifold of dimension at least four. This paper investigates the so called symplectic ball packing problem in the toral equivariant setting. We show that the set of toric symplectic ball packings of M admits the structure of a convex polytope. Previous work of the first author shows that up to equivalence, only CP^1 x CP^1 and CP^2 admit density one packings when n=2 and only CP^n admits density one packings when n>2. In contrast, we show that for a fixed n>=2 and each r in (0, 1), there are uncountably many inequivalent 2n-dimensional symplectic-toric manifolds with a maximal toric packing of density r. This result follows from a general analysis of how the densities of maximal packings change while varying a given symplectic-toric manifold through a family of symplectic-toric manifolds that are equivariantly diffeomorphic but not equivariantly symplectomorphic.