The Symplectic Topology of Ramanujam’s Surface

Ramanujam’s surface M is a contractible affine algebraic surface which is not homeomorphic to the affine plane. For any m > 1 the product M is diffeomorphic to Euclidean space R. We show that, for every m > 0, M cannot be symplectically embedded into a subcritical Stein manifold. This gives the first examples of exotic symplectic structures on Euclidean space which are convex at infinity. It follows that any exhausting plurisubharmonic Morse function on M has at least three critical points, answering a question of Eliashberg. The heart of the argument involves showing a particular Lagrangian torus L inside M cannot be displaced from itself by any Hamiltonian isotopy, via a careful study of pseudoholomorphic discs with boundary on L.