Strict Convexity and Sharp Power Concavity for a Graphical $\sigma_2$-Curvature Equation

Abstract

We prove strict convexity of the square-root transformation $v=-\sqrt{-u}$ for admissible solutions of a graphical $\sigma_2$-curvature Dirichlet problem on smooth uniformly convex domains. A key ingredient is a constant-rank theorem for $D^2v$, proved by a direct Ma-Xu type argument in dimension three and by the Bian-Guan microscopic convexity principle together with inverse-convexity methods in arbitrary dimensions. Combined with boundary strict convexity and a domain-deformation argument, the constant-rank theorem yields $D^2v>0$ throughout the domain.