Fundamental solution of the Laplacian in the hyperboloid model of hyperbolic geometry

Abstract

Due to the isotropy of $d$-dimensional hyperbolic space, one expects there to exist a spherically symmetric fundamental solution for its corresponding Laplace-Beltrami operator. The $R$-radius hyperboloid model of hyperbolic geometry $\Hi_R^d$ with $R>0$, represents a Riemannian manifold with negative-constant sectional curvature. We obtain a spherically symmetric fundamental solution of Laplace's equation on this manifold in terms of its geodesic radius. We give several matching expressions for this fundamental solution including a definite integral over reciprocal powers of the hyperbolic sine, finite summation expression over hyperbolic functions, Gauss hypergeometric functions, and in terms of the associated Legendre function of the second kind with order and degree given by $d/2-1$ with real argument greater than unity. We also demonstrate uniqueness for a fundamental solution of Laplace's equation on this manifold in terms of a vanishing decay at infinity.