Monge-Ampère foliations for degenerate solutions

Abstract We study the problem of the existence and the holomorphicity of the Monge– Ampère foliation associated to a plurisubharmonic solutions of the complex homogeneous Monge–Ampère equation even at points of arbitrary degeneracy. We obtain good results for real analytic unbounded solutions. As a consequence we also provide a positive answer to a question of Burns on homogeneous polynomials whose logarithm satisfies the complex Monge–Ampère equation and we obtain a generalization the work of Wong on the classifi- cation of complete weighted circular domains.