### 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 language | English |
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Pages (from-to) | 727-761 |

Number of pages | 35 |

Journal | Publications of the Research Institute for Mathematical Sciences |

Volume | 43 |

Issue number | 3 |

DOIs | |

Publication status | Published - Sep 1 2007 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

**An expression of harmonic vector fields on hyperbolic 3-cone-manifolds in terms of hypergeometric functions.** / Fujii, Michihiko; Ochiai, Hiroyuki.

Research output: Contribution to journal › Article

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TY - JOUR

T1 - An expression of harmonic vector fields on hyperbolic 3-cone-manifolds in terms of hypergeometric functions

AU - Fujii, Michihiko

AU - Ochiai, Hiroyuki

PY - 2007/9/1

Y1 - 2007/9/1

N2 - 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.

AB - 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.

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U2 - 10.2977/prims/1201012040

DO - 10.2977/prims/1201012040

M3 - Article

VL - 43

SP - 727

EP - 761

JO - Publications of the Research Institute for Mathematical Sciences

JF - Publications of the Research Institute for Mathematical Sciences

SN - 0034-5318

IS - 3

ER -