Minimal Surfaces of Uniplanar Derivation

Abstract

equation (5) becomes equivalent to the Cauchy-Riemann differential equations involving x1 and x2. Consequently, a map of (2) given by an analytic function of a complex variable is a special case of an isothermic map of (2) on a minimal surface. Since, further, minimal surfaces possess many of the properties of maps given by analytic functions, minimal surfaces well might be said to constitute the space analogue of maps given by analytic functions. The purpose of this paper is to define, and to point out some of the properties of, a class of minimal surfaces which seems to correspond closely to that class of maps of (2), given by analytic functions of the complex variable z, which lie on a single-sheeted plane, or, as we shall say, on uniplanar t regions. An equivalent characterization of these functions is that they take on no value more than once in (2). For a discussion of such functions, see, among others, L. Bieberbach, Lehrbuch der Funktionentheorie,