[Paper Review] Fundamental solution of the Laplacian in the hyperboloid model of hyperbolic geometry

Howard S. Cohl, E. G. Kalnins|arXiv (Cornell University)|Jan 20, 2012

Mathematics and Applications19 references4 citations

TL;DR

This paper derives a spherically symmetric fundamental solution (Green's function) for the Laplace-Beltrami operator on the d-dimensional hyperboloid model of hyperbolic space with constant negative curvature. Using geodesic polar coordinates and hyperbolic special functions, the authors present multiple equivalent expressions—via hypergeometric functions, associated Legendre functions of the second kind, and integral representations—proving uniqueness under vanishing decay at infinity.

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.

Motivation & Objective

• To construct a spherically symmetric fundamental solution for the Laplace-Beltrami operator on the R-radius hyperboloid model of hyperbolic space.
• To provide multiple equivalent analytical expressions for this fundamental solution in terms of hyperbolic and special functions.
• To establish uniqueness of the solution under the condition of vanishing decay at infinity.
• To extend the understanding of harmonic analysis on non-compact Riemannian manifolds with negative curvature.
• To unify various representations of the Green's function using integral, series, and special function forms.

Proposed method

• Parametrize the hyperboloid model using geodesic polar coordinates in the Riemannian manifold with constant negative curvature $-1/R^2$.
• Derive radial harmonics in geodesic polar coordinates to construct the fundamental solution as a function of geodesic distance $\rho$.
• Express the fundamental solution via a definite integral involving reciprocal powers of $\sinh\rho$.
• Represent the solution as a finite summation over hyperbolic functions and in terms of Gauss hypergeometric functions.
• Identify the solution with the associated Legendre function of the second kind of order and degree $d/2 - 1$, with real argument $>1$.
• Prove uniqueness by assuming two such solutions and showing their difference must vanish due to the maximum principle and decay condition at infinity.

Experimental results

Research questions

• RQ1 What is the explicit form of the spherically symmetric fundamental solution of the Laplace-Beltrami operator on the d-dimensional hyperboloid model?
• RQ2 How can this fundamental solution be expressed in multiple equivalent mathematical forms, including integrals, series, and special functions?
• RQ3 What conditions ensure the uniqueness of the fundamental solution in hyperbolic space?
• RQ4 How does the decay behavior at infinity affect the uniqueness of the Green's function?
• RQ5 What is the relationship between the fundamental solution and special functions such as the associated Legendre function of the second kind?

Key findings

• The fundamental solution is expressed as a definite integral involving $\sinh^{-d+1}\rho$ terms, providing a direct analytical representation.
• The solution is shown to be equivalent to a finite sum of hyperbolic functions, valid for all $d \geq 2$.
• The solution is represented in terms of the Gauss hypergeometric function $F(a,b;c;z)$, with parameters depending on the dimension $d$.
• The solution is identified as the associated Legendre function of the second kind $Q_{d/2-1}(\cosh\rho)$, with argument $\cosh\rho > 1$.
• The fundamental solution is unique among $C^\infty$ functions that decay to zero at infinity, as proven via the strong maximum principle on non-compact manifolds.
• The solution satisfies $-\Delta \mathcal{H}_R^d = \delta_g$ and decays to zero as geodesic distance $\rho \to \infty$, confirming physical and mathematical consistency.