Symplectic excision and distance rigidity

Abstract

We consider various notions of completeness in symplectic topology and ask two related questions. Does a complete open symplectic manifold remain complete after excising a subset? Can two sets be made arbitrarily far apart by adjusting the almost complex structure within an appropriate class of complete almost complex structures? We find rigidity phenomena when the excised set is a symplectic hypersurface. These arise from certain open Gromov-Witten invariants. We contrast this with flexibility that often occurs when the excised set is coisotropic. We also briefly touch on the opposite question of obstructions to existence of a complete symplectic structure compatible with a given complex structure. For the notion of completeness we first consider the traditional notion of geometric boundedness. We then introduce a broader notion of normalized completeness, related to the notion of intermittent boundedness of \cite{GromanFloerOpen}, which depends on $C^0$ properties and is a contractible condition. Finally we speculate about the relation to a Fukaya-categorical notion of completeness.