2022/08/15 by Wang, Tongrui
#Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2208.07187
Given a compact Riemannian manifold Mn+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.