Min-max theory for free boundary G-invariant minimal hypersurfaces

Abstract

Given a compact Riemannian manifold M n+1 with dimension 3 ≤ n + 1 ≤ 7 and ∂M = ∅, the free boundary min-max theory built by Martin Man-Chun Li and Xin Zhou shows the existence of a smooth almost properly embedded minimal hypersurface with free boundary in ∂M . In this paper, we generalize their constructions into equivariant settings. Specifically, let G be a compact Lie group acting as isometries on M with cohomogeneity at least 3. Then we show that there exists a nontrivial smooth almost properly embedded G-invariant minimal hypersurface with free boundary. Moreover, if the Ricci curvature of M is non-negative and ∂M is strictly convex, then there exist infinitely many properly embedded G-invariant minimal hypersurfaces with free boundary.