JournalscmhVol. 80 , No. 4DOI 10.4171/cmh/37

The symplectic topology of Ramanujam's surface

  • Paul Seidel

    Massachusetts Institute of Technology, Cambridge, USA
  • Ivan Smith

    University of Cambridge, UK
The symplectic topology of Ramanujam's surface cover


Ramanujam's surface MM is a contractible affine algebraic surface which is not homeomorphic to the affine plane. For any m>1m>1 the product MmM^m is diffeomorphic to Euclidean space mathbbR4m{mathbb R}^{4m}. We show that, for every m>0m>0, MmM^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 MmM^m has at least three critical points, answering a question of Eliashberg. The heart of the argument involves showing a particular Lagrangian torus LL inside MM cannot be displaced from itself by any Hamiltonian isotopy, via a careful study of pseudoholomorphic discs with boundary on LL.