Lower estimates for hyperbolic metrics on subdomains

Abstract

A result concerning lower estimates for hyperbolic metrics on subdomains is proved. The method of proof involves a standard technique based on the classical Schwarz–Pick lemma that yields the domain monotonicity of the hyperbolic metric. The information we obtained could be used to estimate, from below, the behavior of the metric near the boundary. We will also investigate the situation when equality holds to show the sharpness of our lower bound.

Keywords:

• hyperbolic metric
• lower estimate
• subdomain
• Schwarz–Pick lemma
• Fuchsian group

AMS Subject Classifications:

• Primary
• 30F45
• 30C80
• Secondary
• 30C25
• 30F35

Acknowledgments

The author is pleased to thank the referees for careful readings of the manuscript and for helpful suggestions.