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 $n\ge4$. 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 $\theta\le1$. 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.