A general conformal geometric reflection principle

Abstract

We prove a generalization of the Schwarz–Carathéodory reflec- tion principle for analytic maps f f from the unit disk into arbitrary Riemann surfaces equipped with a complete real analytic conformal Riemannian metric λ ( w ) | d w | \lambda (w) \, |dw| . This yields a necessary and sufficient condition for f f to have an analytic continuation in terms of the pullback of the metric λ ( w ) | d w | \lambda (w) \, |dw| under the map f f .