RIGIDITY OF CIRCLE DOMAINS WHOSE BOUNDARY HAS SIGMA-FINITE LINEAR MEASURE

Let Q be a circle domain in the Riemann sphere C whose boundary has si gma-finite linear measure. We show that OMEGA is rigid in the sense th at any conformal homeomorphism of Q onto any other circle domain is eq ual to the restriction of a Mobius transformation. Previously, Kaufman and Bishop have independently found examples of non-rigid circle doma ins whose boundary is a Cantor set of (Hausdorff) dimension one.