
 
Closed Strong Spacelike Curves, Fenchel Theorem and Plateau Problem in the 3Dimensional Minkowski Space 
 
Citation： 
Nan YE,Xiang MA.Closed Strong Spacelike Curves, Fenchel Theorem and Plateau Problem in the 3Dimensional Minkowski Space[J].Chinese Annals of Mathematics B,2019,40(2):217~226 
Page view： 48
Net amount： 93 
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 3dimensional 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 2plane 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 Fencheltype
inequality is proved by constructing a ruled spacelike surface with
the given curve as boundary and applying the GaussBonnet formula.
As a byproduct, 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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