Dimension constraints in some problems involving intermediate curvature

TL;DR

This paper investigates dimension constraints for intermediate curvature conditions, providing counterexamples in higher dimensions and extending rigidity results to dimension six, thereby clarifying the sharpness of known theorems.

Contribution

It demonstrates the sharpness of dimension constraints in intermediate curvature problems by constructing counterexamples for dimensions ≥7 and extending rigidity results to dimension 6.

Findings

Counterexamples in dimension ≥7 for intermediate curvature constraints

Rigidity extension to dimension 6 for positive intermediate curvature

Manifolds with dimension ≤5 and bi-Ricci curvature ≥1 have finite Urysohn 1-width

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 $\geq 6$.