Rigidity and flexibility under positive isotropic curvature

Abstract: For every $n\ge4$ and $L&gt;0$, we construct a smooth $4$-PIC metric on $S<sup>n$ with Urysohn $1$-width at least $L$ and an embedded stable minimal disk of intrinsic inradius at least $L$. These examples disprove the proposed width and stable-disk radius bounds under a positive lower bound for isotropic curvature. On closed even-dimensional manifolds, we prove the sharp estimate $λ_1<sup>{(2)}\ge(n-1)σ/2$ under $σ$-PIC and show that equality forces roundness if a closed eigenform attaining the bound has rank at least four at some point.