On the number of solutions to the asymptotic plateau problem

Abstract

By using a simple topological argument, we show that the space of closed, orientable, codimension-1 submanifolds of Sn−1 1 (Hn) which bound a unique absolutely area minimizing hypersurface in Hn is dense in the space of closed, orientable, codimension-1 submanifolds of Sn−1 1 (Hn). In particular, in dimension 3, we prove that the set of simple closed curves in S2 1(H3) bounding a unique absolutely area minimizing surface in H3 is not only dense, but also a countable intersection of open dense subsets of the space of simple closed curves in S2 1(H3) with C0 topology. We also show that the same is true for least area planes in H3. Moreover, we give some non-uniqueness results in dimension 3.