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

TL;DR

This paper extends the Fenchel theorem to strong spacelike closed curves of index 1 in 3D Minkowski space, establishing a curvature bound and linking it to the existence of maximal surfaces with such curves as boundaries.

Contribution

It generalizes the Fenchel theorem to a Minkowski space setting for specific spacelike curves and constructs maximal surfaces bounded by these curves.

Findings

Total curvature of such curves is at most 2π.

Existence of a maximal surface with the given boundary curve.

Construction of a ruled spacelike surface using Gauss-Bonnet formula.

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.