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 R^{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.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 November 2004
 arXiv:
 arXiv:math/0411601
 Bibcode:
 2004math.....11601S
 Keywords:

 Mathematics  Symplectic Geometry;
 Mathematics  Algebraic Geometry;
 53D35;
 14R10
 EPrint:
 22 pages, 2 figures. Version 2 has referee's clarifications, this version to appear in Comment. Math. Helv