The equivariant Plateau problem and interior regularity

Abstract:

Let $M \subset {{\text {R}}^n}$ be a compact submanifold of Euclidean space which is invariant by a compact group $G \subset SO(n)$. When $\dim (M) = n - 2$, it is shown that there always exists a solution to the Plateau problem for $M$ which is invariant by $G$ and, furthermore, that uniqueness of this solution among $G$-invariant currents implies uniqueness in general. This result motivates the subsequent study of the Plateau problem for $M$ within the class of $G$-invariant integral currents. It is shown that this equivariant problem reduces to the study of a corresponding Plateau problem in the orbit space ${\text {R}}/G$ where, for “big” groups, questions of uniqueness and regularity are simplified. The method is then applied to prove that for a constellation of explicit manifolds $M$, the cone $C(M) = \{ tx;x \in M$ and $0 \leqslant t \leqslant 1\}$ is the unique solution to the Plateau problem for $M$, (Thus, there is no hope for general interior regularity of solutions in codimension one.) These manifolds include the original examples of type ${S^n} \times {S^n} \subset {{\text {R}}^{2n + 2}},n \geqslant 3$, due to Bombieri, DeGiorgi, Giusti and Simons. They also include a new example in ${{\text {R}}^8}$ and examples in ${{\text {R}}^n}$ for $n \geqslant 10$ with any prescribed Betti number nonzero.