The Existence of Embedded $G$-Invariant Minimal Hypersurface

Abstract

For a compact connected Lie group $G$ acting as isometries on a compact orientable Riemannian manifold $M^{n+1},$ and cohomogeneity not equal to 0 or 2, 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.

Corrected a mistake pointed out by Antoine Song and added more proofs and discussions at the referee's suggestion. To appear in Calc. Var. Partial Differential Equations