Rigidity of CMC hypersurfaces in 5-and 6-manifolds
Abstract
We prove that nonnegative -intermediate Ricci curvature combined with uniformly positive -triRic curvature implies rigidity of complete noncompact two-sided stable minimal hypersurfaces in a Riemannian manifold with bounded geometry. The stonger assumption of nonnegative -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 -dimensional manifold with positive sectional curvature. This extends result of Chodosh-Li-Stryker [J. Eur. Math. Soc (2025)] to -dimension. We also establish rigidity results on CMC hypersurfaces with nonzero mean curvature in - and -manifolds.
Cite
@article{arxiv.2405.06867,
title = {Rigidity of CMC hypersurfaces in 5-and 6-manifolds},
author = {Han Hong and Zetian Yan},
journal= {arXiv preprint arXiv:2405.06867},
year = {2025}
}
Comments
In this new draft, we refine the statement, improve the argument and extend rigidity results in dimension 6. We also remove the non-existence result on hyperbolic space