Calculus of Variations and Geometric Measure Theory

A. Figalli - A. Pelayo

On the density function on moduli spaces of toric 4-manifolds

created by figalli on 23 Jun 2015


Accepted Paper

Inserted: 23 jun 2015
Last Updated: 23 jun 2015

Journal: Adv. Geom.
Year: 2015


The optimal density function assigns to each symplectic toric manifold $M$ a number $0<d\leq 1$ obtained by considering the ratio between the maximum volume of $M$ which can be filled by symplectically embedded disjoint balls and the total symplectic volume of $M$. In the toric version of this problem, $M$ is toric and the balls need to be embedded respecting the toric action on $M$. The goal of this note is first to give a brief survey of the notion of toric symplectic manifold and the recent constructions of moduli space structure on them, and then recall how to define a natural density function on this moduli space. Then we review previous works which explain how the study of the density function can be reduced to a problem in convex geometry, and use this correspondence to to give a simple description of the regions of continuity of the maximal density function when the dimension is $4$.