Explicit Solving of the System of Natural PDE's of Minimal Space-like Surfaces in Minkowski Space-time

Abstract

A minimal space-like surface in Minkowski space-time is said to be of general type if it is free of degenerate points. The fact that minimal space-like surfaces of general type in Minkowski space-time admit canonical parameters of the first (second) type implies that any minimal space-like surface is determined uniquely up to a motion in space-time by the Gauss curvature and the normal curvature, satisfying a system of two PDE's (the system of natural PDE's). In fact this solves the problem of Lund-Regge for minimal space-like surfaces. Using canonical Weierstrass representations of minimal space-like surfaces of general type in Minkowski space-time we solve explicitly the system of natural PDE's, expressing any solution by means 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 to the system of natural PDE's, i.e. generating one and the same minimal space-like surface in Minkowski space-time.