Area-minimizing integral currents with boundaries invariant under polar actions

Abstract:

Let $G$ be a compact, connected subgroup of $SO(n)$ acting on ${{\mathbf {R}}^n}$, and let the action of $G$ be polar. (Polar actions include the adjoint action of a Lie group $H$ on the tangent space to the symmetric space $G/H$ at the identity coset.) Let $B$ be an $(m - 1)$-dimensional submanifold without boundary, invariant under the action of $G$, and lying in the union of the principal orbits of $G$. It is shown that, if $S$ is an area-minimizing integral current with boundary $B$, then $S$ is invariant under the action of $G$. This result is extended to a larger class of boundaries, and to a class of parametric integrands including the area integrand.