Periodic Reeb orbits on prequantization bundles

TL;DR

This paper proves the existence of at least one or multiple simple periodic Reeb orbits on certain hypersurfaces in prequantization bundles, depending on geometric pinching conditions, extending understanding of Reeb dynamics.

Contribution

It establishes new existence results for periodic Reeb orbits on hypersurfaces in prequantization bundles under specific geometric constraints.

Findings

Existence of at least one simple periodic Reeb orbit under given conditions.

Presence of multiple orbits if cuplength condition is met.

Extension of Reeb orbit existence theory in prequantization bundles.

Abstract

In this paper, we prove that every graphical hypersurface in a prequantization bundle over a symplectic manifold $M$, pinched between two circle bundles whose ratio of radii is less than $\sqrt{2}$ carries either one short simple periodic orbit or carries at least $\operatorname{cuplength}(M)+1$ simple periodic Reeb orbits.