On the Number of Solutions to Asymptotic Plateau Problem

On the Number of Solutions to Asymptotic Plateau Problem

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Baris Coskunuzer, Koc University, Turkey
Fine Hall 401

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 a nonuniqueness result by showing existence of simple closed curves in the sphere at infinity which are the asymptotic boundaries of more than one area minimizing surfaces.