manuscripta mathematica

, Volume 100, Issue 2, pp 123–129

A new variant of the Schwarz–Pick–Ahlfors Lemma

  • Robert Osserman

DOI: 10.1007/s002290050231

Cite this article as:
Osserman, R. manuscripta math. (1999) 100: 123. doi:10.1007/s002290050231


We prove a “general shrinking lemma” that resembles the Schwarz–Pick–Ahlfors Lemma and its many generalizations, but differs in applying to maps of a finite disk into a disk, rather than requiring the domain of the map to be complete. The conclusion is that distances to the origin are all shrunk, and by a limiting procedure we can recover the original Ahlfors Lemma, that all distances are shrunk. The method of proof is also different in that it relates the shrinking of the Schwarz–Pick–Ahlfors-type lemmas to the comparison theorems of Riemannian geometry.

Mathematics Subject Classification (1991):30C80, 53C21 

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Robert Osserman
    • 1
  1. 1.Mathematical Sciences Research Institute, 1000 Centennial Drive, Berkeley, CA 94720-5070, USA.¶e-mail: osserman@msri.orgDE

Personalised recommendations