On the Number of Solutions to Asymptotic Plateau Problem

Details On the Number of Solutions to Asymptotic Plateau Problem On the Number of Solutions to Asymptotic Plateau Problem

2005-05-26

arxiv.org/abs/math/0505593v2

J. Gokova Geom. Topol. 5 (2011) 1-19

Mathematics

Differential Geometry

17 pages, 2 figures. V2: added a new section on nonuniqueness results

Scientific paper

We give a simple topological argument to show that the number of solutions of the asymptotic Plateau problem in hyperbolic space is generically unique. In particular, we show that the space of codimension-1 closed submanifolds of sphere at infinity, which bounds a unique absolutely area minimizing hypersurface in hyperbolic n-space, is dense in the space of all codimension-1 closed submanifolds at infinity. In dimension 3, we also prove that the set of uniqueness curves in asymptotic sphere for area minimizing planes is generic in the set of Jordan curves at infinity. We also give some nonuniqueness results for dimension 3, too.

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