A Classification of Isolated Singularities of Elliptic Monge‐Ampére Equations in Dimension Two

Let ℳ 1 denote the space of solutions z ( x , y ) to an elliptic, real analytic Monge‐Ampére equation det ( D 2 z ) = φ ( x , y , z , D z ) > 0 whose graphs have a non‐removable isolated singularity at the origin. We prove that ℳ 1 is in one‐to‐one correspondence with ℳ 2  × ℤ 2 , where 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‐Ampére 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 + A ( x , y , z , D z ) ) = φ ( x , y , z , D z ) > 0 .© 2015 Wiley Periodicals, Inc.