A new variant of the Schwarz-Pick-Ahlfors lemma

Details A new variant of the Schwarz-Pick-Ahlfors lemma A new variant of the Schwarz-Pick-Ahlfors lemma

1998-03-10

arxiv.org/abs/math/9803158v1

Mathematics

Complex Variables

Scientific paper

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 {\em 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.

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