On the existence of certain general extremal metrics, II

Abstract

1. In applying the method of the extremal metric whether one deals with the problem of extremal length in its customary form [4, 6] or with more general problems [6, 9] the question naturally arises as to the actual existence of an extrernal metric as opposed to the mere consideration of a least upper or greatest lower bound. This is of particular importance in dealing with questions of uniqueness. Actually in his lectures on the subject at Harvard University in 1946-47 Ahlfors showed that in a certain weak sense an extremal metric for an extremal length problem would always exist. However, the metric so obtained would not be certain even to be admissible for the problem and thus it has never proved of any use in applications since for these a fairly high degree of regularity is usually required. Some years ago the author discovered a general principle which makes possible the proof of the existence of the extremal metric for a wide class of problems, in fact precisely those which are most important from the point of view of applications in the Theory of Functions. Reference has been made to this principle in a number of papers [9, 10, 11, 12, 14, 15] and some simple cases have been treated explicitly. In the present paper we will present this principle in its general form. It should be mentioned that the author worked out this problem for the special case of plane domains while at the Institute for Advanced Study in 1949-50 and at that time had the advantage of several conversations with M. I\J. Schiffer on the variational aspects of that case. 2. We consider here finite oriented Riemann surfaces. These are Riemann surfaces of finite genus which may have a finite number of hyperbolic boundary components. Such a surface is conformally equivalent to a domain lying on a closed Riemann surface and bounded by a finite number of analytic curves (possibly zero, so that closed surfaces are included.) On a finite Riemann surface T let there be given a finite number of distinguished points. Let T' be obtained from ? by deleting these points. On 5' we regard homotopy classes of two types. First homotopy classes of simple closed curves in the usual sense Second homotopy classes of arcs which run from one boundary component of T to another (not necessarily distinct) boundary component. In the course of such a homotopy the end points of the arc in question are permitted to move on their respective boundary components. Of course this second type of homotopy class can occur only when T actually has boundary components.