Abstract
Let \(H^n\) be a complete and connected Riemannian manifold with sectional curvature \(K\le -1\) and \(cut(p)\subset H^n\) be the cut locus of a point \( p\in H^n \). We prove that if \( H^n \) contains a closed submanifold, then this submanifold must intersect \( cut(p)\cup \{p\} \) for any p if its mean curvature is not large. As a particular case, we conclude that if a manifold of constant sectional curvature \( -1 \) contains a closed minimal submanifold, then this submanifold must intersect \( cut(p)\cup \{p\} \) for any p of the ambient manifold. This can be viewed as a kind of Frankel property and it explains why there are no closed minimal submanifolds in Hadamard manifolds.
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Meira, A. Cut loci and closed submanifolds of negatively curved manifolds. Arch. Math. 120, 111–113 (2023). https://doi.org/10.1007/s00013-022-01788-0
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DOI: https://doi.org/10.1007/s00013-022-01788-0