Toric symplectic ball packing

TL;DR

This paper characterizes all 2n-dimensional symplectic-toric manifolds that can be perfectly packed with symplectically and torus-equivariantly embedded balls, using combinatorial and convex geometric methods.

Contribution

It introduces a combinatorial problem equivalent to toric symplectic ball packing and solves it using convex geometry and Delzant theory, providing a classification of perfect packings.

Findings

Classified all symplectic-toric manifolds admitting perfect ball packings.

Connected the packing problem to a geometric-combinatorial problem.

Applied results to symplectic blowing-up techniques.

Abstract

We define and solve the toric version of the symplectic ball packing problem, in the sense of listing all 2n-dimensional symplectic-toric manifolds which admit a perfect packing by balls embedded in a symplectic and torus equivariant fashion. In order to do this we first describe a problem in geometric-combinatorics which is equivalent to the toric symplectic ball packing problem. Then we solve this problem using arguments from Convex Geometry and Delzant theory. Applications to symplectic blowing-up are also presented, and some further questions are raised in the last section.