Spacelike minimal surfaces in R^4_1 through of a $\theta$-Family

TL;DR

This paper introduces a $ heta$-family of spacelike surfaces in Lorentz-Minkowski space R^4_1, connecting minimal surfaces in different dimensions, analyzing their curvature, planar points, and providing explicit examples.

Contribution

It defines a new $ heta$-family framework for spacelike surfaces in R^4_1, linking minimal surfaces across dimensions and analyzing their geometric properties.

Findings

The $ heta$-family preserves planar points.

Existence of planar points relates to solutions of |a_w(w)|^2=0.

Certain graph surfaces cannot be locally represented if the imaginary part of a(w) is zero.

Abstract

In this paper we introduce a $\theta$-family of spacelike surfaces in the Lorentz-Minkowski space R^4_1 based in two complex valued functions $a(w), \mu(w)$, which when they are holomorphic we will be dealing with a family of spacelike minimal surfaces. The $\theta$-family is such that it connects spacelike minimal surfaces in R^3_1 to spacelike minimal surfaces in R^3. We study the family through of the curvature and we prove that the family preserves planar points and moreover, that the existence of planar points corresponds to the existence of solutions of equation |a_w(w)|^2 =0. We also show that if a pair of surfaces are associated through of a $\theta$-family then they can not be complete surfaces. As applications we focus to one type of graph surfaces in R^4_1 and we prove that if the imaginary part of $a(w)$ is zero at least in a point then the surface cannot assume local representations of that type of graph. Several explicit examples are given.