Circle packing immersions form regularly exhaustible surfaces^∗

^∗Dedicated to the memory of Glenn Schober.

Abstract

Circle packing imbeddings determine quasiconformai mappings which, in certain situations, approximate conformal mappings. In the more general case of circle packing immersions, it has not been proven that the associated immersion mappings are again quasiregular (i.e., of bounded dilatation but not necessarily univalent). If this were true, then their muitisheeted image surfaces would be regularly exhaustible and consequently Ahlfors' value distribution theory for covering surfaces could be applied to them. In this paper, although we do not settle the question or. whetner such immersions are quasiregular, we are able to prove directly from length-area estimates that the property of being regularly exhaustible holds for the muitisheeted image surfaces. Thus, for example, a Picard type theorem holds: The image of an entire circle packing immersion omits at most one finite value in the complex plane,

AMS No:

• 30A70
• 30A95

^∗Dedicated to the memory of Glenn Schober.

^†Research supported in pat by the N.S.F.

^∗Dedicated to the memory of Glenn Schober.

^†Research supported in pat by the N.S.F.

Notes

^∗Dedicated to the memory of Glenn Schober.

^†Research supported in pat by the N.S.F.