Let M be a complete open 5-manifold with nonnegative Ricci curvature. If its universal cover widetilde M has Euclidean volume growth, meaning that for some widetilde p∈widetilde M one has liminf_r→∞operatornameVol(B_widetilde M(widetilde p,r))/r^5>0, then pi_1(M) is finitely generated and virtually isomorphic to Z^k for some 0≤ k≤ 4.

Five-dimensional Pan–Rong conjecture

Let $M$ be a complete open $5$ -manifold with nonnegative Ricci curvature. If its universal cover $\widetilde M$ has Euclidean volume growth, meaning that for some $\widetilde p\in\widetilde M$ one has $\liminf_{r\to\infty}\operatorname{Vol}(B_{\widetilde M}(\widetilde p,r))/r^5>0$ , then $\pi_1(M)$ is finitely generated and virtually isomorphic to $\mathbb{Z}^k$ for some $0\leq k\leq 4$ .

References

Primary source

arXiv

Additional references

Fundamental Groups in the Five Dimensional Pan-Rong Conjecture — arXiv

Posted 4 weeks ago

Progress summary

Refreshed 4 weeks ago

Claimed solved

An unrefereed preprint claims to settle the conjecture in five dimensions, with an additional quantitative result in six dimensions.

The Pan–Rong conjecture is claimed to hold in dimension five, advancing the structure theory of the relevant open manifolds.

September 2026 claimed settlement

A preprint titled Fundamental Groups in the Five Dimensional Pan-Rong Conjecture claims the conjectured conclusion in dimension five and gives an additional quantitative result in dimension six. The result is unrefereed; higher dimensions remain unresolved.

Current status (as of September 2026): The conjecture is claimed settled in dimension five, with a further six-dimensional result, but the preprint has not been independently verified and higher dimensions remain open.

Sources

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