A New Criterion for Quasiconformal Mapping

Abstract

1. A quasiconformal mapping as originally envisaged by Grdtzsch [4] is in its simplest form a continuously differentiable mapping from a plane domain onto another plane domain or onto a Riemann covering surface. The condition of quasiconformality is then expressed roughly by saying that apart from branch points an infinitesimal circle goes into an infinitesimal ellipse the ratio of whose principal axes is uniformly bounded and uniformly bounded from zero. Grbtzsch himself recognized that the essential properties of quasiconformal mappings were retained under less stringent conditions and Teichmuller indicated the admissibility of quite general exceptional points and curves. It was early realized that almost all proofs of properties of quasiconformal mappings rely on some form of the method of the extremal metric. Consequently, several authors [1, 6] have suggested revising the definition of a quasiconformal mapping, dropping all assumption of differentiability and using the notion of the conformal module of a quadrangle. We will give now this definition confining ourselves for simplicity to homeomorphic sense-preserving mappings from one plane domain to another. However by appropriate use of local uniformizing parameters, our results can be extended to the most general quasiconformal mappings. By a quadrangle Q we mean a simply-connected domain of hyperbolic type with four distinguished boundary elements called vertices. These divide the boundary elements in their natural cyclic order into four sides. The quadrangle admits a conformal mapping onto a rectangle whose pairs of opposite sides have lengths, say a and b, with the vertices of the quadrangle corresponding to the corners of the rectangle. The ratio a/b is a conformal invariant of the quadrangle and it and its reciprocal may be called modules of the quadrangle. They may be obtained directly without reference to the mapping on the rectangle by means of the following module problem. Let r denote the class of open arcs on Q joining a pair of opposite sides of the quadrangle and locally rectifiable in the sense that every closed subarc on them is rectifiable. Suppose that Q lies in the z-plane (z = x + iy) and let p(x, y) be a real valued non-negative func-