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Rigidity of CMC hypersurfaces in 5-and 6-manifolds
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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.
Forward citations
Cited by 2 Pith papers
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Homological $k$-systole in $n$-manifolds with positive intermediate curvature
On closed n-manifolds (n=4,5,6) with gamma-triRic >= n-3 and a nonzero cup product in H^1, the (n-2)-systole is bounded by (n-3)^((n-2)/2)|S^(n-2)| * (inf gamma-triRic)^(-(n-2)/2), with equality only for covers of S^(...
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Complete two-sided $\delta$-stable minimal hypersurfaces in $\mathbf R^{n+1}$
For n=3,4,5 and δ above thresholds δ0(n), complete two-sided δ-stable minimal hypersurfaces in R^{n+1} have Euclidean volume growth, and for δ above δ1(n) they are hyperplanes.
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