Minimal spacelike surfaces and the graphic equations in R^4_1

TL;DR

This paper extends Bernstein's theorem to minimal spacelike surfaces in four-dimensional Minkowski space, characterizing entire solutions of a nonlinear elliptic system and exploring their geometric properties.

Contribution

It introduces a classification of graphic minimal spacelike surfaces in R^4_1, showing the Bernstein property does not hold generally and explicitly constructs conjugated surfaces.

Findings

Bernstein property does not hold for all graphic spacelike surfaces in R^4_1

Classified entire solutions of a nonlinear elliptic system in Minkowski space

Explicitly constructed conjugated minimal spacelike surfaces

Abstract

In this paper we study an extension of the Bernstein Theorem for minimal spacelike surfaces of the four dimensional Minkowski vector space form and we obtain the class of those surfaces which are also graphics and have non-zero Gauss curvature. That is the class of entire solutions of a system of two elliptic non-linear equations that is an extension of the equation of minimal graphic of $\mathbb R^3$. Therefore, we prove that the so-called Bernstein property does not hold in general for the case of graphic spacelike surfaces in $\mathbb R^4_1$. In addition, we also obtain explicitly the conjugated minimal spacelike surface, and identify the necessary conditions to extend continuously a local solution of the generalized Cauchy-Riemann equations.