A new variant of the Schwarz–Pick–Ahlfors Lemma
- Cite this article as:
- Osserman, R. manuscripta math. (1999) 100: 123. doi:10.1007/s002290050231
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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.