A classification of isolated singularities of elliptic Monge-Ampère equations in dimension two

Abstract

Let ${\mathcal{M}}_1$ denote the space of solutions $z(x,y)$ to an elliptic, real analytic Monge-Ampère equation $\det (D^{2}z)=\phi (x,y,z,Dz)>0$ 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-Ampère equation in the $C^k$-smooth case, and a general existence theorem for isolated singularities of analytic solutions of the more general equation $\det (D^{2}z+\mathcal{A}(x,y,z,Dz))=\phi (x,y,z,Dz)>0$.