An Expression of Harmonic Vector Fields on Hyperbolic 3-Cone-Manifolds in Terms of Hypergeometric Functions

Abstract

Let $V$ be a neighborhood of a singular locus of a hyperbolic 3-cone-manifold, which is a quotient space of the 3-dimensional hyperbolic space. In this paper we give an explicit expression of a harmonic vector field $v$ on the hyperbolic manifold $V$ in terms of hypergeometric functions. The expression is obtained by solving a system of ordinary differential equations which is induced by separation of the variables in the vector-valued partial differential equation $(∆+4)\tau =0$, where $∆$ is the Laplacian of $V$ and $\tau$ is the dual 1-form of $v$. We transform this system of ordinary differential equations to single-component differential equations by elimination of unknown functions and solve these equations. The most important step in solving them consists of two parts, decomposing their differential operators into differential operators of the type appearing in Riemann’s $P$-equation in the ring of differential operators and then describing the projections to the components of this decomposition in terms of differential operators that are also of the type appearing in Riemann’s $P$-equation.