Given a complete weighted Riemannian manifold (M,g,e^-f) with bounded potential f and nonnegative Bakry–Émery Ricci curvature, namely operatornameRic_g+operatornameHess_g f≥ 0, must the underlying manifold M admit a Riemannian metric h such that operatornameRic_h≥ 0?

Wei–Wylie question on Bakry–Émery curvature and Ricci curvature

Wei and Wylie asked in 2007 whether a complete manifold with bounded weighted potential and nonnegative Bakry–Émery curvature must admit some metric with nonnegative ordinary Ricci curvature.

Known results

Wei–Wylie, 2007: established comparison and volume-comparison theorems under lower bounds on $\operatorname{Ric}_f$ with bounded $f$ or suitable radial derivative bounds; Question 7.5 remained unanswered there.

August 2026 claimed negative answer

The preprint Positive Bakry–Émery Ricci Curvature on Homotopy Spheres reports weighted core metrics on homotopy spheres whose nonzero $\alpha$ -invariants obstruct nonnegative-Ricci metrics, giving a negative answer in dimensions $8k+1$ and $8k+2$ for $k \ge 1$ . The supplied searches provide no independent verification or corroboration of this claim.

Current status (as of August 2026): A negative answer is claimed but unverified in dimensions $8k+1$ and $8k+2$ for $k \ge 1$ ; the question remains open in the other dimensions.

Wei–Wylie question on Bakry–Émery curvature and Ricci curvature

Primary source

Additional references

Progress summary

Solutions 0