Elliptic Curves and Algebraic Geometry Approach in Gravity Theory III. Uniformization Functions for a Multivariable Cubic Algebraic Equation

TL;DR

This paper extends the algebraic and geometric analysis of multivariable cubic equations in gravity theory, demonstrating that certain parametrization functions serve as uniformization functions depending on complex variables, with applications to metrics like ADS space.

Contribution

It shows that parametrization functions in gravity can be viewed as uniformization functions of complex variables, extending to multiple variables for physical metric analysis.

Findings

Parametrization functions are also uniformization functions.

Uniformization extended to two complex variables.

Application to ADS metric of constant negative curvature.

Abstract

The third part of the present paper continues the investigation of the solution of the multivariable cubic algebraic equation for reparametrization invariance of the gravitational Lagrangian. The main result in this paper constitutes the fact that the earlier found parametrization functions of the cubic algebraic equation for reparametrization invariance of the gravitational Lagrangian can be considered also as uniformization functions. These functions are obtained as solutions of first - order nonlinear differential equations, as a result of which they depend only on the complex (uniformization) variable z. Further, it has been demonstrated that this uniformization can be extended to two complex variables, which is particularly important for investigating various physical metrics, for example the ADS metric of constant negative curvature (Lobachevsky spaces).