Volume-preserving transformations in several complex variables

Abstract

In this paper we study pseudo-conformal mappings, that is, transformations of four-dimensional continua by means of pairs of analytic functions, u(z1, Z2), V(Z1, Z2), of two complex variables z1, Z2. In particular we study volume-preserving transformations, that is, mappings in which the volume of any four-dimensional domain D in the Z1z2 space equals the volume of the domain D1 into which D is carried by the mapping (zI, Z2)-4-(U, v). As in the case of mappings by means of one complex variable, we adopt the convention that the volumes of regions which are covered by mappings more than once are counted with their multiplicities. It may be pointed out that the investigation of these maps constitutes a genuine several problem, since the corresponding maps associated with one complex variable reduce to trivial translations and rotations. A particularly interesting subclass of the class of volume-preserving transformations is the class of mappings which are also univalent, that is, mappings which are one-to-one in both directions. Some families of such mappings will be exhibited. The study of volume-preserving pseudo-conformal maps is suggested by the work of Bergman and others [1 ]2 on pseudo-conformal maps of given domains which minimize the volume under certain conditions. Since the solutions of such minimum problems are determined only up to an arbitrary volume-preserving mapping, the investigation of the latter may prove to be useful in arriving at certain unique extremal mappings characterized by geometric properties. It is easy to see that the volume V1 of the pseudo-conformal image D1 of D is [1, p. 138]