Strict Convexity for Solution of Liouville-Type Dirichlet Problems

We identify a common convexity structure for three exponential Dirichlet problems on smooth uniformly strictly convex domains: the Liouville equation $\mathrm{\Delta }u=e^u$, the real equation ${\sigma }_2(D^{2}u)=e^{2u}$, and its complex counterpart ${\sigma }_2(u_{i\bar{j}})=e^{2u}$. In each case $u<0$ in the domain and $u=0$ on the boundary. We prove that $$w=-\mathrm{arcosh}(e^{-u\mathrm{/}2})$$ is strictly convex in the underlying real variables. The argument combines domain deformation, constant-rank theory, inverse-convexity estimates, radial ball models, boundary strict convexity, and local $C^2$ stability.