Canonical coordinates and natural equations for minimal time-like surfaces in $\mathbf{R}^4_2$

Abstract

We apply the complex analysis over the double numbers $\mathbf{D}$ to study the minimal time-like surfaces in $\mathbf{R}^4_2$. 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 $\varkappa$, satisfying the system of natural equations for these surfaces. We prove the following: If $(K, \varkappa),\, K^2- \varkappa^2 > 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 $(K, \varkappa)$; if $(K, \varkappa),\, K^2- \varkappa^2$ < 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 $(K, \varkappa)$.