The geometry of poincaré disks

Abstract

A simply connected planar domain Ω of finite area is said to be a Poincare disk if there exists a finite positive constant K such that

for all functions u which are C ¹ on Ω and integrate to zero. In this paper we establish geometric necessary and sufficient conditions for Ω to be a Poincarè disk. Our criteria, which are reminiscent of the isoperimetric inequality and simplify a characterization of Maz'ja, state that the smaller area determined from a crosscut of Ω must be bounded by a constant multiple of the length of the crosscut. We show that this characterization is valid for three different types of crosscuts: line segments, hyperbolic geodesies, and general crosscuts. We also obtain a characterization of those conformal mappings which map the disk onto a Poincarè disk in terms of an integral growth condition. We use techniques from, geometric function theory and hyperbolic geometry.

AMS No:

• 30C20
• 46E35
• 51M10