Invariant solutions to the oriented Plateau problem of maximal codimension

Abstract:

The principal result gives conditions which imply that a solution to the Plateau problem inherits the symmetries of its boundary. Specifically, let G be a compact connected Lie subgroup of ${\text {SO}}(n)$. Assume the principal orbits have dimension m, there are no exceptional orbits and the distribution of $(n - m)$-planes orthogonal to the principal orbits is involutive. We show that if B is a finite sum of oriented principal orbits, then every absolutely area minimizing current with boundary B is invariant. As a consequence of the methods used, the above Plateau problems are shown to be equivalent to 1-dimensional variational problems in the orbit space. Some results concerning invariant area minimizing currents in Riemannian manifolds are also obtained.