Automorphic Forms and Reeb-Like Foliations on Three-Manifolds

Abstract

In this paper, we consider different ways of generating dynamical systems on 3-manifolds. We first derive explicit differential equations for dynamical systems defined on generic hyperbolic 3-manifolds by using automorphic function theory to uniformize the upper half-space model. It is achieved via the modification of the standard Poincare theta series to generate systems invariant within each individual fundamental region such that the solution trajectories match up on the appropriate sides after the identifications which generate a hyperbolic 3-manifold. Then we consider the gluing pattern in the conformal ball model. At the end we shall study the construction of dynamical systems by using the Reeb foliation.