Rigidity of CMC hypersurfaces in 5-and 6-manifolds

Han Hong; Zetian Yan

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arXiv · 2405.06867

Rigidity of CMC hypersurfaces in 5-and 6-manifolds

Han Hong·Zetian Yan

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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.

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Han Hong, Zetian Yan. 2024-05-11. Rigidity of CMC hypersurfaces in 5-and 6-manifolds. https://arxiv.org/abs/2405.06867

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