Equivariant Morse index of min–max G-invariant minimal hypersurfaces

For a closed Riemannian manifold M n + 1 with a compact Lie group G acting as isometries, the equivariant min–max theory gives the existence and the potential abundance of minimal G -invariant hypersurfaces provided 3 ≤ codim ( G · p ) ≤ 7 for all p ∈ M . In this paper, we show a compactness theorem for these min–max minimal G -hypersurfaces and construct a G -invariant Jacobi field on the limit. Combining with an equivariant bumpy metrics theorem, we obtain a C G ∞ -generic finiteness result for min–max G -hypersurfaces with area uniformly bounded. As a main application, we further generalize the Morse index estimates for min–max minimal hypersurfaces to the equivariant setting. Namely, the closed G -invariant minimal hypersurface Σ ⊂ M constructed by the equivariant min–max on a k -dimensional homotopy class can be chosen to satisfy Index G ( Σ ) ≤ k .