Shen’s strict conjugate-radius conjecture

Shen’s conjecture predicts a strict conjugate-radius bound below $\pi$ in dimension three under a scalar-curvature hypothesis.

Known results

• Green (1963): compact manifolds with scalar curvature at least $n(n-1)$ have conjugate radius at most $\pi$, with equality only for the round sphere.
• Zhu (2022): an open-manifold result under nonnegative Ricci curvature.
• Kwong (2025): a finite-volume result.
• A related open-manifold theorem gives $\operatorname{conj}(M)\le\pi\sqrt{1-\lambda_0(M)/n}$ when $\operatorname{scal}_M\ge n(n-1)$ and $\lambda_0(M)<n$; in particular, $\operatorname{conj}(M)\le\pi$ when $\lambda_0(M)=0$.

October 2026 claimed resolution

Jian Ge and Chuanhuan Li’s preprint A conjugate radius bound for open three-manifolds claims a universal $2\pi/3$ bound, which would imply Shen’s strict inequality below $\pi$ in dimension three. The claim is not independently assessed.

Current status (as of October 2026): A preprint claims the conjecture follows from a universal $2\pi/3$ bound, but the claimed resolution remains unverified.