Contractible Closed Reeb Orbits on Prequantization Bundles under Weak Index Conditions

In this paper we study the multiplicity of contractible closed Reeb orbits on prequantization circle bundles. Building on the existing multiplicity result Theorem 1.3 for closed Reeb orbits on prequantization $S^1$-bundles, we weaken the index assumptions by requiring only that the supporting contact form has no contractible periodic Reeb orbits with mean Conley-Zehnder index zero and no contractible good periodic Reeb orbits with Conley-Zehnder index 0, and we obtain a lower bound for the number of geometrically distinct contractible closed Reeb orbits carried by contact forms $\alpha$ on a prequantization $S^1$-bundle under certain natural topological assumptions (Theorem 1.6). In dimension five we further relax the index assumptions and show that, under a mild index condition imposed only on simple contractible periodic orbits and the same topological assumptions on the prequantization $S^1$-bundle, every non-degenerate supporting contact form carries at least $\dim H_{\ast }(B;\mathbb{Q})$ geometrically distinct contractible closed Reeb orbits (Theorem 1.11), in agreement with the conjectural lower bound for prequantization bundles.