Closed Strong Spacelike Curves, Fenchel Theorem and Plateau Problem in the 3-Dimensional Minkowski Space

Citation:

Nan YE,Xiang MA.Closed Strong Spacelike Curves, Fenchel Theorem and Plateau Problem in the 3-Dimensional Minkowski Space[J].Chinese Annals of Mathematics B,2019,40(2):217~226
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Authors:

Nan YE; Xiang MA

Foundation:

This work was supported by the National Natural Science Foundation of China (No.11471021) and the Fundamental Research Funds for the Central Universities of China (No.531107050874).
Abstract: The authors 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 the strong spacelike condition 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. This reversed Fenchel-type inequality is proved 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 the boundary.

Keywords:

Fenchel theorem, Spacelike curves, Total curvature, Maximal surface

Classification:

52A40, 53C42, 53C50
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