Israel Journal of Mathematics

, Volume 30, Issue 1–2, pp 152–158 | Cite as

On Teichmüller’s theorem on the quasi-invariance of cross ratios

  • Irwin Kra


Teichmüller’s theorem gives necessary and sufficient conditions for mapping one ordered quadruple by aK-quasiconformal map onto a second ordered quadruple. We give a simple non-computational proof of the necessity part. We then characterize such extremal mappings, and obtain as a consequence a new formula for the modular function, with leads to a very simple derivation of the known expression for the Poincaré metric on the thrice-punctured sphere.


Quasiconformal Mapping Quadratic Differential Extremal Mapping Cross Ratio Beltrami Coefficient 
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Copyright information

© Hebrew University 1978

Authors and Affiliations

  • Irwin Kra
    • 1
  1. 1.State University of New York at Stony BrookStony BrookUSA

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