The Ahlfors lemma and Picard's theorems
Aleksander Simoni\v{c}
TL;DR
This paper discusses Ahlfors' generalization of the Schwarz lemma and its applications to Picard's theorems, providing insights into the size of images under holomorphic maps and Kobayashi hyperbolic manifolds.
Contribution
It introduces Ahlfors' generalized Schwarz lemma and demonstrates its use in proving Picard's theorems and exploring complex hyperbolic geometry.
Findings
Ahlfors' lemma extends Schwarz's lemma to broader contexts.
Holomorphic mappings have constrained image sizes due to Ahlfors' lemma.
Insights into Kobayashi hyperbolic manifolds are provided.
Abstract
The article introduces Ahlfors' generalization of the Schwarz lemma. With this powerful geometric tool of complex functions in one variable, we are able to prove some theorems concerning the size of images under holomorphic mappings, including the celebrated Picard's theorems. The article concludes with a brief insight into the theory of Kobayashi hyperbolic complex manifolds.
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Taxonomy
TopicsHolomorphic and Operator Theory · Geometry and complex manifolds · Geometric and Algebraic Topology