Gromov’s width and stable-minimal-disk conjectures under positive isotropic curvature

For each $n\ge 4$ and each $\sigma>0$ , there should exist a constant $C(n,\sigma)<\infty$ such that every smooth closed $n$ -manifold with isotropic curvature bounded below by $\sigma$ has Urysohn $1$ -width at most $C(n,\sigma)$ , and every embedded stable minimal disk in it has intrinsic inradius at most $C(n,\sigma)$ .

References

Primary source

arXiv

Additional references

Rigidity and flexibility under positive isotropic curvature — arXiv — Tsz-Kiu Aaron Chow, Yipeng Wang

minimal-surfaces riemannian-geometry

Posted 3 weeks ago · edited 3 weeks ago

Progress summary

Refreshed 3 weeks ago

Claimed solved

A September 2026 preprint claims that the proposed bounds fail on smooth positively isotropic-curved four-dimensional spheres, while a separate conjecture is untouched.

Gromov’s width and stable-minimal-disk conjectures predict geometric bounds under positive isotropic curvature. The latest report claims both bounds fail, without addressing the separate virtual-freeness conjecture.

September 10, 2026 counterexample

Tsz-Kiu Aaron Chow and Yipeng Wang report counterexamples showing that the proposed width and stable-disk bounds fail even for smooth $4$ -PIC metrics on spheres. This is a claimed refutation of the tracked conjectures, but the available evidence does not independently verify the result.

Current status (as of September 2026): The width and stable-minimal-disk conjectures are claimed false for smooth $4$ -PIC metrics on spheres, but the claim is unverified; the separate virtual-freeness conjecture is not addressed.

Sources

No solutions have been posted yet.