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| Closed Strong Spacelike Curves, Fenchel Theorem and Plateau Problem in the 3-Dimensional Minkowski Space |
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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 |
| Page view： 1604
Net amount： 1665 |
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). |
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| 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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