Rigidity and nonexistence of CMC hypersurfaces in 5-manifolds

We prove that the nonnegative 3-intermediate Ricci curvature and uniformly positive k-triRic curvature implies rigidity of complete noncompact two-sided stable minimal hypersurfaces in a Riemannian manifold (X⁵, g) with bounded geometry. The nonnegativity of 3-intermediate Ricci curvature can be replaced by nonnegative Ricci and biRic curvature. In particular, there is no complete noncompact finite index CMC hypersurface in a closed 5-dimensional manifold with positive sectional curvature. It extends result of Chodosh-Li-Stryker to appear in J. Eur. Math. Soc (2024) to 5-dimensions. We also prove that complete constant mean curvature hypersurfaces in hyperbolic space H⁵ with finite index and the mean curvature greater than 658 must be compact. This improves the previous larger bound 175148 on the mean curvature.