Rigidity and flexibility under positive isotropic curvature

Abstract

For every $n\ge 4$ and $L>0$, we construct a smooth $4$-PIC metric on $S^n$ 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 ${\lambda }_1^{(2)}\ge (n-1)\sigma /2$ under $\sigma$-PIC and show that equality forces roundness if a closed eigenform attaining the bound has rank at least four at some point.

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