By using a simple topological argument, we show that the space of closed, orientable, codimension-1 submanifolds of S^n-1_∞(Hⁿ) which bound a unique absolutely area minimizing hypersurface in Hⁿ is dense in the space of closed, orientable, codimension-1 submanifolds of S^n-1_∞(Hⁿ). In particular, in dimension 3, we prove that the set of simple closed curves in S²_∞(Hⁿ) 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_∞(Hⁿ) 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.