Hopf product conjecture

The conjecture asks whether $\mathbb{S}^2 \times \mathbb{S}^2$ admits a Riemannian metric with positive sectional curvature. It was popularized in 1968 and listed as Problem 1 in Yau’s problems.

Known results

• Bourguignon ruled out positive curvature near the product metric.
• Hopf ruled out an embedding of a positively curved example in $\mathbb{R}^5$.
• Weinstein ruled out such an embedding in $\mathbb{R}^6$.
• Hsiang and Kleiner ruled out positively curved metrics with an isometric circle action.

August 2026 claimed counterexample

Brendle and Hung’s preprint claims that a third-order perturbation of a Cheeger–Müter metric yields positive sectional curvature on $\mathbb{S}^2 \times \mathbb{S}^2$. If correct, this refutes the Hopf product conjecture. The preprint is unrefereed, its abstract does not name the conjecture, and no independent verification was found.

Current status (as of August 2026): The conjecture is not yet established as false, but Brendle and Hung’s unrefereed preprint claims a counterexample; independent verification remains open.