Brin’s quarter-pinching conjecture

Brin’s conjecture concerns whether strict quarter-pinching forces ergodicity of the oriented frame flow on closed negatively curved manifolds. The new work addresses this implication only in selected dimensions and geometric regimes.

Known results

• Brin and Gromov proved ergodicity without pinching in every odd dimension $n\ne 7$.

10 August 2026 partial progress

Heng Zhang and Shuhao Zhang report that strict $\frac14$-pinching implies ergodicity for $n=4$ and for $n\equiv 2\pmod{4}$ with $n\ne 134$. They also obtain ergodicity from strict $\frac{5}{13}$-pinching when $4\mid n$ and $n\ge 12$, plus improved thresholds in dimensions $7$, $8$, and $134$. This is an unrefereed preprint and explicitly does not resolve all dimensions.

Current status (as of August 2026): Several dimensional cases are claimed in an unrefereed preprint, while Brin’s quarter-pinching conjecture remains open in full generality.