EQUIVARIANT MIN-MAX HYPERSURFACE IN G-MANIFOLDS WITH POSITIVE RICCI CURVATURE

We consider a connected orientable closed Riemannian manifold M n + 1 with positive Ricci curvature. Suppose G is a compact Lie group acting by isometries on M with 3 <= codim(G <middle dot> p) <= 7 for all p is an element of M. Then we show the equivariant min-max G-hypersurface E corresponding to one- parameter G-sweepouts (of boundary-type) is a multiplicity one minimal G-hypersurface with a G-invariant unit normal and G-equivariant index one. As an application, we are able to establish a genus bound for E, a control on the singular points of E/ G, and an upper bound for the (first) G-width of M provided n + 1 = 3 and the actions of G are orientation preserving.