Finite type Monge–Ampère foliations

Abstract

For plurisubharmonic solutions of the complex homogeneous Monge–Ampère equation whose level sets are hypersurfaces of finite type, in dimension 2, it is shown that the Monge–Ampère foliation is defined even at points of higher degeneracy. The result is applied to provide a positive answer to a question of Burns on homogeneous polynomials whose logarithms satisfy the complex Monge–Ampère equation and to generalize the work of P. M. Wong on the classification of complete weighted circular domains.

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Advances in Geometry is a mathematical journal which publishes original research articles of excellent quality in the area of geometry. Geometry is a field of long-standing tradition and eminent importance. The study of space and spatial patterns is a major mathematical activity, and geometric ideas and the geometric language permeate all of mathematics.

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