Closed strong spacelike curves, Fenchel theorem and Plateau problem in the 3-dimensional Minkowski space

We generalize the Fenchel theorem for strong spacelike closed curves of index $1$ in the 3-dimensional Minkowski space, showing that the total curvature must be less than or equal to $2π$. Here strong spacelike means that the tangent vector and the curvature vector span a spacelike 2-plane at each point of the curve $γ$ under consideration. The assumption of index 1 is equivalent to saying that $γ$ winds around some timelike axis with winding number 1. We prove this reversed Fenchel-type inequality by constructing a ruled spacelike surface with the given curve as boundary and applying the Gauss-Bonnet formula. As a by-product, this shows the existence of a maximal surface with $γ$ as boundary.

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