Canonical coordinates and natural equations for minimal time-like surfaces in R⁴₂

Abstract

We apply the complex analysis over the double numbers D to study the minimal time-like surfaces in . A minimal time-like surface which is free of degenerate points is said to be of general type. We divide the minimal time-like surfaces of general type into three types and prove that these surfaces admit special geometric (canonical) parameters. Then the geometry of the minimal time-like surfaces of general type is determined by the Gauss curvature K and the curvature of the normal connection κ, satisfying the system of natural equations for these surfaces. We prove the following: If (K, κ), K²-κ² > 0 is a solution to the system of natural equations, then there exists exactly one minimal time-like surface of the first type and exactly one minimal time-like surface of the second type with invariants (K, κ); if (K, κ), K²-κ² < 0 is a solution to the system of natural equations, then there exists exactly one minimal time-like surface of the third type with invariants (K, κ).