A new variant of the Schwarz-Pick-Ahlfors lemma
Complex Variables
2016-09-07 v1
Abstract
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.
Cite
@article{arxiv.math/9803158,
title = {A new variant of the Schwarz-Pick-Ahlfors lemma},
author = {Robert Osserman},
journal= {arXiv preprint arXiv:math/9803158},
year = {2016}
}