Boundaries of open symplectic manifolds and the failure of packing stability

Abstract

A finite volume symplectic manifold is said to have “packing stability” if the only obstruction to symplectically embedding sufficiently small balls is the volume obstruction. Packing stability has been shown in a variety of cases and it has been conjectured that it always holds. We give counterexamples to this conjecture; in fact, we give examples that cannot be fully packed by any domain with smooth boundary nor by any convex domain. Our examples are symplectomorphic to open and bounded domains in ${\mathbb{R}}^4$, with the diffeomorphism type of a ball. The obstruction to packing stability is closely tied to another old question, which asks to what extent an open symplectic manifold has a well-defined boundary; it follows from our results that many examples cannot be symplectomorphic to the interior of any compact symplectic manifold with smooth boundary. Our results can be quantified in terms of the volume decay near the boundary, and we produce, for example, smooth toric domains that are only symplectomorphic to the interior of a compact domain if the boundary of this domain has inner Minkowski dimension arbitrarily close to $4$. The growth rate of the subleading asymptotics of the ECH spectrum plays a key role in our arguments. We prove a very general “fractal Weyl law”, relating this growth rate to the Minkowski dimension; this formula is of potentially independent interest.