The modulus of a doubly connected region and the geodesic curvature-area method

C. David (Carl) Minda

Vol. 113 (1984), No. 2, 395–406

DOI: 10.2140/pjm.1984.113.395

Abstract

It is well known that the modulus of a doubly connected Riemann surface can be determined by the length-area method, that is, the method of extremal length, and that the extremal metric can be expressed in terms of a quadratic differential. Ahlfors introduced a related method based on the comparison of geodesic curvature and area. We show that the modulus of a doubly connected Riemann surface can be obtained by means of this geodesic curvature-area method. In the important special case in which there is a restriction on the curvature of the metrics, we identify all extremal metrics; they have constant curvature.

Mathematical Subject Classification 2000

Primary: 30F30

Secondary: 53A05

Milestones

Received: 1 November 1982

Published: 1 August 1984