Commentarii Mathematici Helvetici


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Volume 80, Issue 4, 2005, pp. 859–881
DOI: 10.4171/CMH/37

Published online: 2005-12-31

The symplectic topology of Ramanujam's surface

Paul Seidel[1] and Ivan Smith[2]

(1) Massachusetts Institute of Technology, Cambridge, USA
(2) University of Cambridge, UK

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 ${mathbb R}^{4m}$. 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 $L$.

Keywords: Stein manifold, contractible affine surface, exotic symplectic structure, subcritical Stein manifold

Seidel Paul, Smith Ivan: The symplectic topology of Ramanujam's surface. Comment. Math. Helv. 80 (2005), 859-881. doi: 10.4171/CMH/37