Discontinuous symplectic capacities

TL;DR

This paper demonstrates the discontinuity of certain symplectic capacities, specifically the spherical and shell capacities, on smooth families of ellipsoidal shells and open sets with smooth connected boundaries.

Contribution

It establishes the first known examples of discontinuity of symplectic capacities on smooth families of sets.

Findings

Spherical capacity is discontinuous on ellipsoidal shells.

Shell capacity is discontinuous on open sets with smooth boundaries.

Provides new insights into the behavior of symplectic invariants.

Abstract

We show that the spherical capacity is discontinuous on a smooth family of ellipsoidal shells. Moreover, we prove that the shell capacity is discontinuous on a family of open sets with smooth connected boundaries.