Canonical Coordinates and Natural Equations for Minimal Time-like Surfaces in $R^4_2$

TL;DR

This paper studies minimal time-like surfaces in four-dimensional pseudo-Euclidean space using complex analysis over double numbers, classifies them into three types, and establishes unique correspondence with solutions of natural equations based on their invariants.

Contribution

It introduces a classification of minimal time-like surfaces of general type into three types and proves their correspondence with solutions of natural equations via canonical parameters.

Findings

Surfaces are classified into three types based on invariants.

Existence and uniqueness of surfaces are established for each type.

Natural equations determine the geometry of these surfaces.

Abstract

We apply the complex analysis over the double numbers $D$ to study the minimal time-like surfaces in $R^4_2$. 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 $(K, \varkappa)$.