A sharp threshold for mixed $Q$-curvature rigidity

Abstract

Let $I_a(g)=Q_g+aσ_2(A_g)$, where $A_g$ is the Schouten tensor and $Q_g$ is Branson's $Q$-curvature. On a closed connected manifold of dimension $n\ge4$ with a positive Einstein metric $g_0$, we prove that every smooth metric conformal to $g_0$ with nonnegative scalar curvature and constant $I_a(g)$ is Einstein for $a\ge-4$. This lower threshold is sharp in every dimension: for each sufficiently small $η>0$, the round sphere admits a smooth non-Einstein conformal metric with positive scalar curvature and constant $I_{-4-η}(g)$. The rigidity proof combines the pointwise Obata identity with a Newton identity whose reference curvature is the minimum of the scalar curvature. A local pole equation and integral estimates yield a radial shooting construction joining a neck to nearly round caps. The rescaled necks converge to the Riemannian Schwarzschild metric, and we determine the asymptotic neck scale. We also prove rigidity for $I_a(g)=ΛR_g^θ$ when $R_g>0$, $a\ge-4$, and $θ\le1$. In dimension four, constant-$I_a(g)$ rigidity holds for $-4\le a\le-4/3$ without a scalar-curvature sign assumption.