The Interpolation Problem for Schlicht Functions

Abstract

circle onto the points wo, w1, ... , w , by a function w = f(z) which is analytic and schlicht (univalent) in the unit circle. In the present paper we consider not only this problem but also the refinement of it which arises by fixing the topological type of f. The topological type of f can be defined in two different ways and the variational techniques applied to this problem show that they are equivalent. This results in an identification of homotopy and conformal deformation classes which is reminiscent of the work of Morse and Heins [8]. Using variational techniques [9, 11, 12, 13] we characterize extremal functions by a hyperelliptic differential equation. From the uniqueness of the schlicht functions satisfying these differential equations we derive the topological result mentioned above and are able to associate to each set (w0, .-- , wn) and topological type a canonical function f in such a way that the correspondence which associates to f the point I(f) to which it belongs is a homeomorphism. For the case of three points we give an explicit criterion for the possibility of such a mapping. This criterion involves elliptic and hyperelliptic moduli.