The existence of embedded G-invariant minimal hypersurface

Abstract

For a compact connected Lie group G acting as isometries of cohomogeneity not equal to 0 or 2 on a compact orientable Riemannian manifold \(M^{n+1},\) we prove the existence of a nontrivial embedded G-invariant minimal hypersurface, that is smooth outside a set of Hausdorff dimension at most \(n-7.\)