Dimension constraints in some problems involving intermediate curvature

In [Comm. Pure Appl. Math. 77 (2024), pp. 441–456] Brendle-Hirsch-Johne proved that T m × S n − m T^m× Sⁿ⁻ᵐ does not admit metrics with positive m m -intermediate curvature when n ≤ 7 n≤ 7 . Chu-Kwong-Lee showed in [Math. Res. Lett., to appear] a corresponding rigidity statement when n ≤ 5 n≤ 5 . In this paper, we show sharpness of the dimension constraints by giving concrete counterexamples in n ≥ 7 n≥ 7 and extending the rigidity result to n = 6 n=6 . Concerning uniformly positive intermediate curvature, we show that simply-connected manifolds with dimension ≤ 5 ≤ 5 and bi-Ricci curvature ≥ 1 ≥ 1 have finite Urysohn 1-width. Counterexamples are constructed in dimension ≥ 6 ≥ 6 .