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arXiv · 2301.02730

Dimension constraints in some problems involving intermediate curvature

Kai Xu
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Abstract

In arXiv:2207.08617 [math.DG] Brendle-Hirsch-Johne proved that $T^m\times S^{n-m}$ does not admit metrics with positive $m$-intermediate curvature when $n\leq 7$. Chu-Kwong-Lee showed in arXiv:2208.12240 [math.DG] a corresponding rigidity statement when $n\leq 5$. In this paper, we show the sharpness of the dimension constraints by giving concrete counterexamples in $n\geq 7$ and extending the rigidity result to $n=6$. Concerning uniformly positive intermediate curvature, we show that simply-connected manifolds with dimension $\leq 5$ and bi-Ricci curvature $\geq 1$ have finite Urysohn 1-width. Counterexamples are constructed in dimension $\geq 6$.

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Kai Xu. 2023-01-06. Dimension constraints in some problems involving intermediate curvature. https://arxiv.org/abs/2301.02730

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