# Canonical Weierstrass Representations for Maximal Space-like Surfaces in $\RR^4_2$

## TL;DR

This paper introduces a canonical Weierstrass representation for maximal space-like surfaces in four-dimensional pseudo-Euclidean space, enabling explicit solutions to the natural PDEs via holomorphic functions and revealing geometric correspondences.

## Contribution

It develops a new canonical Weierstrass representation for these surfaces, linking solutions of PDEs to pairs of holomorphic functions and establishing geometric correspondences.

## Key findings

- Explicit Weierstrass formulas for maximal space-like surfaces.
- Solution of natural PDEs using holomorphic functions.
- Correspondence between surfaces in different pseudo-Euclidean spaces.

## Abstract

It is known that any maximal space-like surface without isotropic points in the four-dimensional pseudo-Euclidean space with neutral metric admits locally geometric parameters which are special case of isothermal parameters. With respect to such parameters the surface is determined uniquely up to a motion by the Gauss curvature and the curvature of the normal connection, which satisfy a system of two PDE's (the system of natural PDE's). For any maximal space-like surface parametrized by canonical parameters we obtain a special Weierstrass representation -- canonical Weierstrass representation. These Weierstrass formulas allow us to solve explicitly the system of natural PDE's by virtue of two holomorphic functions in the Gauss plane. We find the relation between two pairs of holomorphic functions generating one and the same solution to the system of natural PDE's. We establish a geometric correspondence between the maximal space-like surfaces of general type in $\RR^4_2$, the solutions to the system of natural PDE's and the pairs of holomorphic functions in the Gauss plane. We prove that any maximal space-like surface in the four-dimensional pseudo-Euclidean space with neutral metric generates two maximal space-like surfaces in the three-dimensional Minkowski space and vice versa.