On the Klein-Gordon Equation on Some Examples of Conformally Flat Spin 3-Manifolds

In this paper we present an overview about our recent results on the analytic treatment of the Klein-Gordon equation on some conformally flat 3-tori and on 3-spheres. In the first part of this paper we consider the time independent Klein-Gordon equation (Δ−α 2)u=0 (α∈ℝ) on some conformally flat 3-tori associated with a representative system of conformally inequivalent spinor bundles. We set up an explicit formula for the fundamental solution associated to each spinor bundle. We show that we can represent any solution to the homogeneous Klein-Gordon equation on such a torus as finite sum over generalized 3-fold periodic or resp. antiperiodic elliptic functions that are in the kernel of the Klein-Gordon operator. Furthermore, we prove Cauchy and Green type integral formulas and set up an appropriate Teodorescu and Cauchy transform for the toroidal Klein-Gordon operator on this spin tori. These in turn are used to set up explicit formulas for the solution to the inhomogeneous Klein-Gordon equation (Δ−α 2)u=f on the 3-torus attached to the different choices of different spinor bundles. In the second part of the paper we present a unified approach to describe the solutions to the Klein-Gordon equation on 3-spheres. We give an explicit representation formula for the solutions in terms of hypergeometric functions and monogenic homogeneous polynomials.