||Journal of Gökova Geometry Topology, Volume 5 (2011)|
||On the number of solutions to the asymptotic Plateau problem|
By using a simple topological argument, we show that the space of closed, orientable, codimension-1 submanifolds of Sn-1∞(Hn) which bound a unique
absolutely area minimizing hypersurface in Hn is dense in the space of closed, orientable, codimension-1 submanifolds of
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.
||Asymptotic Plateau problem, uniqueness, hyperbolic space, least area plane, absolutely area minimizing surface|
|Submitted: ||July 4, 2010|
|Accepted: ||Nov 20, 2011|
2011 Journal main page|
Last updated: December 2011
Web address: GokovaGT.org/journal/2011