The Plateau problem for polygonal boundary curves in Minkowski 3-space

Details The Plateau problem for polygonal boundary curves in Minkowski 3-space The Plateau problem for polygonal boundary curves in Minkowski 3-space

2010-12-16

arxiv.org/abs/1012.3582v1

Mathematics

Differential Geometry

32 pages

Scientific paper

We apply Garnier's method to solve the Plateau problem for maximal surfaces in Minkowski 3-space. Our study relies on the improved version we gave of R. Garnier's resolution of the Plateau problem for polygonal boundary curves in Euclidean 3-space. Since in Minkowski space the method does not allow us to avoid the existence of singularities, the appropriate framework is to consider maxfaces -- generalized maximal surfaces without branch points, introduced by M. Umehara and K. Yamada. We prove that any given spacelike polygonal curve in generic position in Minkowski 3-space bounds at least one maxface of disk-type. This is a new result, since the only known result for the Plateau problem in Minkowski space (due to N. Quien) deals with boundary curves of regularity $C^{3,\alpha}$.

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