Symplectic homology. I: Open sets in \(\mathbb{C}^ n\) (Q1323418)

The purpose of this paper is to construct a symplectic homology on \(\mathbb{C}^ n\) by using Floer homology and capacity theory. This theory of the symplectic homology has many important applications. As an example, the computation of the symplectic homology groups of polydisks leads to a proof of a conjecture of Gromov on the symplectic classification of open polydisks: For \(r = (r_ 1, \dots, r_ n)\) with the convention \(0 < r_ 1 \leq r_ 2 \leq \dots \leq r_ n\), let \(B^ 2(r_ i)\) be an open boule of radius \(r_ i\) in \(\mathbb{C}\) and \(D(r)\) be the polydisk \(B^ 2(r_ 1) \times \dots \times B^ 2(r_ n)\). Then \(D(r)\) and \(D(r')\) are symplectomorphic if and only if \(r = r'\).