The symplectic topology of Ramanujam's surface

Abstract

Ramanujam's surface $M$ is a contractible affine algebraic surface which is not homeomorphic to the affine plane. For any $m>1$ the product $M^m$ is diffeomorphic to Euclidean space ${mathbbR}^{4m}$. We show that, for every $m>0$, $M^m$ 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 $M^m$ 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 $L$.