Closed Strong Spacelike Curves, Fenchel Theorem and Plateau Problem in the 3-Dimensional Minkowski Space
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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 2π. 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 γ under consideration. The assumption of index 1 is equivalent to saying that γ 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 γ as the boundary.
KeywordsFenchel theorem Spacelike curves Total curvature Maximal surface
2010 MR Subject Classification52A40 53C42 53C50
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The authors are grateful for discussing with Donghao Wang on various constructions of the desired saddle shaped surface with given boundary γ and on the gradient estimation problem. The second author would like to thank Keomkyo Seo for introducing the paper  and motivating his interest in this topic. The authors are also thankful for the suggestions of the referee helping them to clarify several points and improve the presentation of this paper. The authors are grateful to Dinghong Liu for pointing out a gap in the original argument of Section 4.
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