A Comparison Pogorelov Estimate for Graphical ${\sigma }_k$-Curvature Equations

We establish comparison Pogorelov estimates for admissible solutions of graphical ${\sigma }_k$-curvature equations, extending the two-surface estimate of Qiu and Yan for the graphical scalar curvature equation. The comparison graph is assumed only to be $k$-admissible, with a bounded slope and a positive interior gap that vanishes on the boundary. The estimates cover $2\le k<n$ for prescribed data $f(x,u)$, and $n\mathrm{/}2\le k<n$ for general data $f(x,u,Du)$. The proof combines the two-surface localization and mixed Gårding comparison of Qiu and Yan with Yan’s spectral concavity inequality and an angle-function weight.