JournalscmhVol. 80, No. 4pp. 859–881

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
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Abstract

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.

Cite this article

Paul Seidel, Ivan Smith, The symplectic topology of Ramanujam's surface. Comment. Math. Helv. 80 (2005), no. 4, pp. 859–881

DOI 10.4171/CMH/37