Rigidity of CMC hypersurfaces in 5-and 6-manifolds

TL;DR

This paper establishes rigidity results for stable minimal and CMC hypersurfaces in 5- and 6-dimensional manifolds under certain curvature conditions, extending previous results to higher dimensions.

Contribution

It extends rigidity theorems for stable minimal hypersurfaces to 5-dimensional manifolds with specific curvature conditions and establishes new results for CMC hypersurfaces in 5- and 6-manifolds.

Findings

No complete noncompact stable minimal hypersurface in a 5-manifold with positive sectional curvature.

Rigidity results for CMC hypersurfaces with nonzero mean curvature in 5- and 6-manifolds.

Curvature conditions imply hypersurface rigidity in higher dimensions.

Abstract

We prove that nonnegative $3$-intermediate Ricci curvature combined with uniformly positive $k$-triRic curvature implies rigidity of complete noncompact two-sided stable minimal hypersurfaces in a Riemannian manifold $(X^5,g)$ with bounded geometry. The stonger assumption of nonnegative $3$-intermediate Ricci curvature can be replaced by the nonnegativity of Ricci and biRic curvature. In particular, there is no complete noncompact stable minimal hypersurface in a closed $5$-dimensional manifold with positive sectional curvature. This extends result of Chodosh-Li-Stryker [J. Eur. Math. Soc (2025)] to $5$-dimension. We also establish rigidity results on CMC hypersurfaces with nonzero mean curvature in $5$- and $6$-manifolds.