On the Number of Solutions to Asymptotic Plateau Problem

Abstract

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.

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

Cited by in corpus (4)

• Asymptotic Plateau Problem
• Number of Least Area Planes in Gromov Hyperbolic 3-Spaces
• Minimizing Constant Mean Curvature Hypersurfaces in Hyperbolic Space
• Generic non-uniqueness of complete $H$-surfaces embedded in ${\mathbb{H}}^3$