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

Abstract

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 $2\pi$. Here strong spacelike means that the tangent vector and the curvature vector span a spacelike 2-plane at each point of the curve $\gamma$ under consideration. The assumption of index 1 is equivalent to saying that $\gamma$ 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 $\gamma$ as boundary.