Rigidity, sharp inequalities, and stability for $σ_2$-curvature by a reference curvature technique

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 $σ_2$-flat conformal hemispheres with $A_g\in\overline{Γ_2^+}$ and positive constant natural boundary curvature. (ii) On every closed po