On the Number of Solutions to Asymptotic Plateau Problem

TL;DR

This paper demonstrates that in hyperbolic space, the solutions to the asymptotic Plateau problem are generically unique, with the set of boundary conditions leading to unique solutions being dense, and explores conditions for nonuniqueness.

Contribution

It introduces a topological argument showing generic uniqueness of solutions and characterizes the density of boundary conditions with unique solutions in hyperbolic space.

Findings

Solutions are generically unique in hyperbolic space.

The set of boundary conditions with unique solutions is dense.

Nonuniqueness occurs under certain conditions in dimension 3.

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.