A Theorem Concerning the Existence of Deformable Conformal Maps

Abstract

1. We shall say that a Riemann surface F admits deformable conformal maps into itself (or belongs to the class ?2) provided that there exists a continuous map f from F x [0,1] into F satisfying the two conditions: (a) for each t e [0,1], f, : p -? f(p, t) is a (directly) conformal map (not necessarily univalent) of F into itself; (b) fo # f1 . A problem which presents itself at once in the study of the homotopy of conformal maps is to characterize the Riemann surfaces of class ?. It is easy to see that the sphere, the finite plane, the punctured plane and the tori all belong to the class ?r. There remain to be considered the Riemann surfaces with hyperbolic universal covering surface. These may be divided into two classes: one being the class of hyperbolic Riemann surfaces, the other the class of Riemann surfaces which themselves are not hyperbolic but which have hyperbolic universal covering surface. No surface of this latter class can belong to 2~. The case of compact Riemann surfaces of genus > 2 is classical. There are only a finite number of conformal maps of such a surface into itself. The validity of the assertion for the case of noncompact surfaces is established in [3; ?? 6, 7]. Thus there remain for consideration only the hyperbolic Riemann surfaces. In this paper we shall establish the following THEOREM. A hyperbolic Riemann surface belongs to the class ?Y if and only if it admits non-constant bounded analytic functions. That the existence of non-constant bounded analytic functions is a sufficient condition for a Riemann surface to belong to the class ?2! is of course trivial. We shall therefore confine our attention to showing that this condition is necessary for hyperbolic surfaces. The proof will be based on our prior studies on the Lindelof principle [1] and Lindelofian maps [2]. These papers will be referred to henceforth as LP and LM respectively. 2. We recall [LM] that a non-constant meromorphic function qb whose domain is a hyperbolic Riemann surface F is termed Lindeldfian provided that