Symplectic Excision and Distance Rigidity

A symplectic manifold $M$ is called tame at infinity if it admits a compatible almost complex structure such that the corresponding Riemannian metric is complete and geometrically bounded. Some such condition is necessary to confine $J$-holomorphic curves of finite symplectic area. In fact, the strict geometric boundedness condition can be relaxed to a weakly contractible condition that still allows for the same confinement.

Because there is no distinguished such almost complex structure, we ask: Are there geometric features common to all of them? We investigate this through the lens of distances between subsets of $M$. A non-quantitative version of the same question is: does $M$ remain tame upon excising a subset? We find rigidity phenomena when excising symplectic hypersurfaces, which contrast with the flexibility that often occurs when the excised set is coisotropic.