Symplectic cohomology of contractible surfaces

Abstract

In 2004, Seidel and Smith proved that the Liouville manifold associated to Ramanujams surface contains a Lagrangian torus which is not displaceable by Hamiltonian isotopy, and hence that higher products of this manifold provide non-standard symplectic structures on Euclidean space which are convex at infinity. I extend these techniques a wide class of smooth contractible affine surfaces of log-general type to produce a similar torus. I then show that the existence of this torus implies the non-vanishing of the symplectic cohomology of the Liouville manifolds associated to these surfaces, thus confirming a portion of McLeans conjecture that a smooth variety has vanishing symplectic cohomology if and only if it is affine ruled.