Rigidity, sharp inequalities, and stability for $σ_2$-curvature

Using classical conformal divergence identities with a variable reference curvature, we prove four main results. First, we establish a general mean-curvature estimate for conformal metrics on the round hemisphere with $A_g\in\overline{Γ_2^+}$ and positive prescribed $H_2$ data. When $σ_2(A_g)=0$ and