Uniform domains on holomorphic curves

Abstract

We study a recent criterion for the injectivity of a holomorphic curve in , the strong form of which is shown to imply that the image in is a uniform surface. This concept is borrowed from geometric function theory in connection with quasidisks. We also obtain a two-point distortion theorem giving sharp estimates for the separation of images in terms of the hyperbolic distance in .

Keywords:

• Schwarzian derivative
• univalence
• uniform domain
• holomorphic curve

AMS Subject Classifications:

• 30C99
• 30C55
• 31A35
• 53A05

Acknowledgements

This research is Partially supported by the Universidad Nacional de Colombia, Hermes code 21080 and Fondecyt grant 1110321. The author wishes to thank the University of Cauca for providing time for this work through research project VRI ID 3979.