Explicit Solving of the System of Natural PDE's of Minimal Surfaces in the Four-Dimensional Euclidean Space

Abstract

The fact that minimal surfaces in the four-dimensional Euclidean space admit natural parameters implies that any minimal surface is determined uniquely up to a motion by two curvature functions, satisfying a system of two PDE's (the system of natural PDE's). In fact this solves the problem of Lund-Regge for minimal surfaces. Using the corresponding result for minimal surfaces in the three-dimensional Euclidean space, we solve explicitly the system of natural PDE's, expressing any solution by virtue of two holomorphic functions in the Gauss plane. We find the relation between two pairs of holomorphic functions (i.e. the class of pairs of holomorphic functions) generating one and the same solution of the system of natural PDE's.