Marginally trapped surfaces in ${\mathbb{L}}^4$ and three Weierstrass representations

Abstract

We construct new integrable systems to present Weierstrass type representations for spacelike surfaces whose mean curvature vector $\mathbf{H}$ satisfies the null condition $⟨\mathbf{H},\mathbf{H}⟩=0$ in the four dimensional Lorentz-Minkowski space ${\mathbb{L}}^4$. Our new Weierstrass presentations extend simultaneously classical Weierstrass representations (of the first kind and the second kind) for maximal surfaces in ${\mathbb{L}}^3$ and minimal surfaces in ${\mathbb{R}}^3$. We solve a linear partial differental equation to construct explicit examples of marginally trapped surfaces with nowhere vanishing mean curvature vector.

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