The equivariant plateau problem and interior regularity

Let M CR” be a compact submanifold of Euclidean space which is invariant by a compact group G C SO (n). When dim (A/) = n — 2, it is shown that there always exists a solution to the Plateau problem for M which is invari® ant by G and, furthermore, that uniqueness of this solution among G®invariant cur® rents 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 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 e M and 0 < t < l! 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 in® elude the original examples of type Sⁿ x Sⁿ C R^^w+^, n > 3, due to Bombieri, DeGiorgi, Giusti and Simons. They also include a new example in R^ and examples in R^w for n > 10 with any prescribed Betti number nonzero.