A sharp threshold for mixed $Q$-curvature rigidity

Abstract

We resolve Case's question (Crelle's Journal, 2024) on the sharp lower threshold for Obata-type rigidity of $I_a=Q+a{\sigma }_2(A)$ in positive Einstein conformal classes of dimension $n\ge 4$. Using a new reference-curvature identity, we prove that, on a closed connected manifold, every smooth metric in such a class with nonnegative scalar curvature and constant $I_a$ is Einstein whenever $a\ge -4$. The scalar-curvature sign assumption can be removed for $-4\le a\le -2(n-2)/(n-1)$. We also extend the quotient rigidity theorem of Ge--Wang--Wei to $I_a=\Lambda R^{\theta }$ for $R>0$, $a\ge -4$, and $\theta \le 1$. Sharpness is established in every dimension by a shooting construction. For each sufficiently small $\eta >0$, we construct a smooth non-Einstein metric in the round conformal class on ${\mathbb{S}}^n$ with positive scalar curvature and the same constant $I_{-4-\eta }$ as the unit round metric. The construction smoothly matches a perturbed Schwarzschild neck near the equator with perturbed round caps at both poles.

arXiv comment29 pages. Revised version. Rigidity without a scalar-curvature sign assumption is extended to all dimensions $n\ge 4$ for $-4\le a\le -2(n-2)/(n-1)$. The proof of the shooting construction has been revised and clarified, and the references have been updated