Regular domains and surfaces of constant Gaussian curvature in \(3\)-dimensional affine space (Q2152576)

This paper studies a so-called constant affine Gaussian curvature (CAGC) equation \[ \begin{cases} \det(D^2u)=c_k(-w_\Omega)^{-n-2}, & \mbox{ in } U\equiv\mbox{int dom}(u)\subset\Omega,\\ u\Big|_{\partial\Omega}=\varphi,\\ |\nabla u(x)|\to+\infty, & \mbox{ as } x\in U \mbox{ tends to } \partial U, \end{cases} \] on \({\mathbb{R}}^{n}\), which unifies two equations \[ \begin{cases} \det(D^2w)=(-w)^{-n-2}, & \mbox{ in } \Omega,\\ w\Big|_{\partial\Omega}=0, \end{cases} \] and \[\begin{cases} \det(D^2u)=(1-|x|)^{-2}, & \mbox{ in } U\equiv\mbox{int dom}(u),\\ u\Big|_{\partial{\mathbb{D}}}=\varphi,\\ |\nabla u(x)|\to+\infty, & \mbox{ as } x\in U \mbox{ tends to }\partial U \end{cases} \] for a constant \(c_k>0\) determined by \(k\) and \(n\). In the three-dimensional case \(n+1=3\), the authors show that every proper regular domain is uniquely foliated by some particular surfaces with constant affine Gaussian curvature. Their argument is based on an analysis of a Monge-Ampère equation with extended real-valued lower semicontinuous boundary condition.