Invariance of solutions to invariant parametric variational problems

Abstract:

Let G be a compact Lie group of diffeomorphisms of a connected orientable manifold M of dimension $n + 1$. Assume the orbits of highest dimension to be connected. Let $\Psi$ be a convex positive even parametric integrand of degree n on M which is invariant under the action of G. Let T be a homologically $\Psi$-minimizing rectifiable current of dimension n on M, and assume there exists a G-invariant rectifiable current $T’$ which is homologous to T. It is shown that T is G-invariant provided $\Psi$ satisfies a symmetry condition which makes it no less efficient for the tangent planes of T to lie along the orbits. This condition is satisfied by the area integrand in case G is a group of isometries of a Riemannian metric on M. Consequently, one obtains the corollary that if a rectifiable current T is a solution to the n-dimensional Plateau problem in M with G-invariant boundary $\partial T$, and if $\partial T$ bounds a G-invariant rectifiable current $T’$ such that $T - T’$ is a boundary, then T is G-invariant. An application to the Plateau problem in ${{\textbf {S}}^3}$ is given.