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

Shuning Xu

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arXiv · 2608.20907

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

Shuning Xu

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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.

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Shuning Xu. 2026-08-21. Strict Convexity and Sharp Power Concavity for a Graphical $\sigma_2$-Curvature Equation. https://arxiv.org/abs/2608.20907

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