Curvy points, the perimeter, and the complexity of convex toric domains

TL;DR

This paper investigates the geometric and symplectic properties of convex toric domains, revealing how curvature and perimeter influence symplectic packing, and classifies domains with infinite staircases based on boundary curvature.

Contribution

It introduces new results linking boundary curvature to symplectic invariants and classifies convex toric domains with infinite staircases, extending several theorems to generalized settings.

Findings

Perimeter recovered from ECH capacities without genericity.

Failure of packing stability in certain manifolds.

Curvature at boundary points obstructs infinite staircases.

Abstract

We study the related notions of curvature and perimeter for toric boundaries and their implications for symplectic packing problems; a natural setting for this is a generalized version of convex toric domain which we also study, where there are no conditions on the moment polytope at all aside from convexity. We show that the subleading asymptotics of the ECH and elementary ECH capacities recover the perimeter of such domains in their liminf, without any genericity required, and hence the perimeter is an obstruction to a full filling. As an application, we give the first examples of the failure of packing stability by open subsets of compact manifolds with smooth boundary or with no boundary at all; this has implications for long-term super-recurrence. We also show that a single smooth point of positive curvature on the toric boundary obstructs the existence of an infinite staircase, and we build on this to completely classify smooth (generalized) convex toric domains which have an infinite staircase. We also extend a number of theorems to generalized convex toric domains, in particular the "concave to convex", embedding theorem and the "accumulation point theorem". A curvy point forces "infinite complexity"; we raise the question of whether an infinitely complex domain can ever have an infinite staircase and we give examples with infinite staircases and arbitrarily high finite complexity.