The minimum points of the hyperbolic metric^∗

^∗Dedicated to the memory of our friend Glenn Schober.

Abstract

We consider two main problems. гirst, what are the properties of the set E(Ω) of points where the density λ_ω(z) of the hyperbolic metric on a hyperbolic plane domain Ω attains its absolute minimum. Second, what are the properties of the subset D(Ω) (D _S (Ω)) of a domain Ω where all holomorphic (meromorphic) self-mappings of Ω are distance-decreasing relative to euclidean (spherical) geometry. For a simply connected proper subdomain of the plane, the two sets E(Ω) and D(Ω) coincide, while on general hyperbolic domains, the second contains the first, and may strictly contain it. The set D(Ω) can be empty, while D _s (Ω) is always nonempty. We give conditions both for the existence of minimum points and for the discreteness of the set of minimum points. We show that minimum points of the hyperbolic density cannot get too close to the boundary in a uniform sense.

AMS No:

• 30F45
• 30D45

^∗Dedicated to the memory of our friend Glenn Schober.

^∗∗Research partially supported by NSF Grant No. DMS-9008051.

^∗Dedicated to the memory of our friend Glenn Schober.

^∗∗Research partially supported by NSF Grant No. DMS-9008051.

Notes

^∗Dedicated to the memory of our friend Glenn Schober.

^∗∗Research partially supported by NSF Grant No. DMS-9008051.