Toshiyuki SUGAWA

Various domain constants related to uniform perfectness


This is a survey article on domain constants related to uniform perfectness. We gather comparison theorems for various domain constants, most of which are, more or less, known or elementary, but not stated quantitatively in literature, and some are new or improved results.
Among these theorems, our main result is a comparison of the modulus and the injectivity radius of a hyperbolic Riemann surface. Its proof relies upon a comparison of extremal and hyperbolic lengths, which seems to be interesting in itself.
And we include a lower estimate of the Hausdorff dimension of a compact set in the Riemann sphere by the modulus of its complement. We also discuss the variance of these domain constants under conformal, quasiconformal or M\"obius maps.

submission: June 23, 1997
revised: July 31, 1997

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