Rigidity, sharp inequalities, and stability for $\sigma_2$-curvature by a reference curvature technique

Abstract

Using a reference-curvature technique, we prove four main theorems. (i) For $n\geq4$, we remove the mean-curvature pinching from the Case-Wang rigidity theorem for $\sigma_2$-flat conformal hemispheres with $A_g\in\overline{\Gamma_2^+}$ and positive constant natural boundary curvature. (ii) On every closed positive Einstein manifold of dimension $n\geq5$, each conformal metric of constant $\sigma_2$-curvature is Einstein, and is homothetic to the background unless the background is a round sphere, thereby resolving the $\sigma_2$-Obata problem posed by Chang, Gursky, and Yang. (iii) We extend the spherical constant-$\sigma_2/\sigma_1$ classification and sharp integral comparison to closed positive Einstein manifolds. The comparison holds throughout the positive-scalar-curvature conformal class, with equality only for homotheties on nonround backgrounds. (iv) We establish global quantitative $\sigma_2$ stability on every fixed nonround positive Einstein background of dimension $n\geq5$, throughout its positive-scalar-curvature conformal class.