# A Cauchy problem for minimal spacelike surfaces in $\mathbb{R}^4_2$

## TL;DR

This paper defines isoclinic parametric surfaces in a pseudo-Euclidean space, proves their relation to holomorphic functions, and solves a Cauchy problem for constructing minimal spacelike surfaces containing a given curve, also exploring the Björling problem.

## Contribution

It introduces a new class of isoclinic surfaces in $ ext{R}^4_2$, establishes their holomorphic characterization, and solves the Cauchy problem for minimal spacelike surfaces with prescribed boundary curves.

## Key findings

- Isoclinic conformal immersions are generated by two holomorphic functions.
- The Cauchy problem for minimal spacelike surfaces is solvable in this setting.
- Examples of solutions to the Björling problem are provided.

## Abstract

We give a definition of isoclinic parametric surfaces in $\mathbb{R}^4_2$ and prove that such an isoclinic conformal immersion comes from two holomorphic functions. A Cauchy problem was proposed and solved, namely: construct an isoclinic and minimal positive (negative) spacelike surface in $\mathbb{R}^4_2$, containing a given positive (negative) real analytic curve. At last, we study the important and well-known Bj\"orling problem, providing some examples was given in the last section.