**A. Pereira do Vale & M. R. Pinto (ed.),
**

Proceedings of the 1st International Meeting on Geometry and Topology.

Proceedings of the conference held in Braga, September 11-13, 1997.

Universidade do Minho, Portugal

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A new variant of the Schwarz-Pick-Ahlfors Lemma

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Robert Osserman

MSRI, 1000 Centennial Drive, Berkeley, CA 94720-5070, USA osserman@msri.org

**Abstract:** We prove a "general shrinking lemma" that resembles the Schwarz-Pick-Ahlfors Lemma and its 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 the proof is also different in that relates the shrinking of the Schwarz-Pick-Ahlfors-type lemmas to the comparison theorems of Riemannian geometry.

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