### Abstract

For any prescribed differential principal part on a compact Riemann surface, there are uniquely determined and intrinsically defined meromorphic abelian differentials with these principal parts, defined independently of any choice of a marking of the surface or of a basis for the holomorphic abelian differentials, and they are holomorphic functions of the singularities. They can be constructed explicitly in terms of intrinsically defined cross-ratio functions on the Riemann surfaces, the classical cross-ratio function for the Riemann sphere, and natural generalizations for surfaces of higher genus.

Original language | English (US) |
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Title of host publication | From Fourier Analysis and Number Theory to Radon Transforms and Geometry |

Subtitle of host publication | In Memory of Leon Ehrenpreis |

Editors | Hershel Farkas, Marvin Knopp, Robert Gunning, B.A Taylor |

Pages | 303-324 |

Number of pages | 22 |

DOIs | |

State | Published - Sep 2 2013 |

### Publication series

Name | Developments in Mathematics |
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Volume | 28 |

ISSN (Print) | 1389-2177 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

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## Cite this

Gunning, R. C. (2013). Some intrinsic constructions on compact Riemann surfaces. In H. Farkas, M. Knopp, R. Gunning, & B. A. Taylor (Eds.),

*From Fourier Analysis and Number Theory to Radon Transforms and Geometry: In Memory of Leon Ehrenpreis*(pp. 303-324). (Developments in Mathematics; Vol. 28). https://doi.org/10.1007/978-1-4614-4075-8_13