An expression of harmonic vector fields on hyperbolic 3-cone-manifolds in terms of hypergeometric functions

Michihiko Fujii, Hiroyuki Ochiai

Research output: Contribution to journalArticle

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 (Δ + 4)τ = 0, where Δ is the Laplacian of V and τ 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.

Original languageEnglish
Pages (from-to)727-761
Number of pages35
JournalPublications of the Research Institute for Mathematical Sciences
Volume43
Issue number3
DOIs
Publication statusPublished - Sep 1 2007

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Hypergeometric Functions
Differential operator
Vector Field
Cone
Harmonic
System of Ordinary Differential Equations
Ring of Differential Operators
Hyperbolic Manifold
Quotient Space
Hyperbolic Space
Locus
Elimination
Partial differential equation
Projection
Transform
Differential equation
Decompose
Unknown

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

Cite this

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