Multiplicity of closed Reeb orbits on contact manifolds with periodic equivariant symplectic homology

Abstract

We consider closed contact manifolds $(M,\xi )$ with periodic positive equivariant symplectic homology. This is a very large class of contact manifolds and, to the best of our knowledge, includes all currently known examples admitting Reeb flows with finitely many closed orbits for which this homology is well defined. Under weak and homologically natural index assumptions on a non-degenerate contact form $\alpha$ on $M$, we establish a sharp lower bound $r_M$ for the number of simple closed Reeb orbits of $\alpha$. The quantity $r_M$ is completely defined in terms of the positive equivariant symplectic homology of $M$. Moreover, we show that this bound is attained if and only if $\alpha$ is lacunary, i.e., the Conley-Zehnder indices of all closed orbits have the same parity. Consequently, the invariant $r_M$ admits a clean dynamical characterization: whenever $M$ admits a non-degenerate lacunary contact form, $r_M$ equals the number of its simple closed Reeb orbits and is therefore independent of the choice of such a form. In particular, in the lacunary case the number of such orbits is a contact invariant completely determined by Floer theory. We compute $r_M$ for a broad class of examples, including several prequantizations of symplectic orbifolds, and show that in this case $r_M=\dim H_{\ast }(M/S^1;\mathbb{Q})$, thereby giving a topological characterization of the invariant. Motivated by these results, we conjecture that any contact form with finitely many closed Reeb orbits is necessarily non-degenerate and lacunary, and that the underlying contact manifold is a prequantization of this type.

72 pages. Version 2: minor changes