A classification of isolated singularities of elliptic Monge-Amp\`ere equations in dimension two

Abstract

Let $\mathcal{M}_1$ denote the space of solutions $z(x,y)$ to an elliptic, real analytic Monge-Amp\`ere equation ${\rm det} (D^2 z)=\varphi(x,y,z,Dz)>0$ whose graphs have a non-removable isolated singularity at the origin. We prove that $\mathcal{M}_1$ is in one-to-one correspondence with $\mathcal{M}_2\times Z_2$, where $\mathcal{M}_2$ is a suitable subset of the class of regular, real analytic strictly convex Jordan curves in $R^2$. We also describe the asymptotic behavior of solutions of the Monge-Amp\`ere equation in the $C^k$-smooth case, and a general existence theorem for isolated singularities of analytic solutions of the more general equation ${\rm det} (D^2 z +\mathcal{A}(x,y,z,Dz))=\varphi(x,y,z,Dz)>0$.