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

Abstract

Given a compact Riemannian manifold Mn+1 with dimension 3≤n+1≤7 and ∂M≠∅, the free boundary min-max theory built by Li-Zhou [19] 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 there exists a nontrivial smooth almost properly embedded free boundary G-invariant minimal hypersurface. 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.