Some intrinsic constructions on compact Riemann surfaces

Robert C. Gunning

Research output: Chapter in Book/Report/Conference proceedingChapter


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 languageEnglish (US)
Title of host publicationFrom Fourier Analysis and Number Theory to Radon Transforms and Geometry
Subtitle of host publicationIn Memory of Leon Ehrenpreis
EditorsHershel Farkas, Marvin Knopp, Robert Gunning, B.A Taylor
Number of pages22
StatePublished - 2013

Publication series

NameDevelopments in Mathematics
ISSN (Print)1389-2177

All Science Journal Classification (ASJC) codes

  • General Mathematics


  • Abelian differentials
  • Cross-ratio function
  • Riemann surfaces


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