Packing stability and the subleading asymptotics of symplectic Weyl laws

Abstract

We prove that symplectic ball packing stability holds for every compact, connected symplectic $4$-manifold with smooth boundary. This follows from a stronger result: the full volume of any such manifold can be filled by a single symplectic ellipsoid. As an application, we obtain estimates - with sharp exponents - for the error terms in the symplectic Weyl laws for embedded contact homology capacities, periodic Floer homology spectral invariants, and link spectral invariants. We also construct an example of a star-shaped domain in ${\mathbb{R}}^4$, arbitrarily $C^1$ close to the unit ball and with boundary of regularity just below $C^2$ and smooth away from a single point, for which packing stability fails. Our proofs reveal a close connection between symplectic packing stability in the presence of smooth boundary and the algebraic structure of Hamiltonian diffeomorphism groups, particularly Banyaga's simplicity results.

62 pages, 21 figures