Rigidity and nonexistence of CMC hypersurfaces in 5-manifolds
Authors: Han Hong, Zetian Yan
Summary: We show that the nonnegative 3-intermediate Ricci curvature and uniformly constructive k-triRic curvature implies rigidity of full noncompact two-sided steady minimal hypersurfaces in a Riemannian manifold (X5,g) with bounded geometry. The nonnegativity of 3-intermediate Ricci curvature might be changed by nonnegative Ricci and biRic curvature. Specifically, there isn’t any full noncompact finite index CMC hypersurface in a closed 5-dimensional manifold with constructive sectional curvature. It extends results of Chodosh-Li-Stryker [to appear in J. Eur. Math. Soc (2024)] to 5-dimensions. We additionally show that full fixed imply curvature hypersurfaces in hyperbolic house H5 with finite index and the imply curvature larger than 65√8 should be compact. This improves the earlier bigger sure 175√148√ on the imply curvature.