Monge-Ampere foliations for degenerate solutions

TL;DR

This paper investigates the existence and holomorphicity of Monge-Ampère foliations for degenerate plurisubharmonic solutions, providing new insights into their structure and answering longstanding questions in complex analysis.

Contribution

It establishes conditions for the holomorphicity of Monge-Ampère foliations at degeneracy points and generalizes previous classifications of certain complex domains.

Findings

Proves holomorphicity of foliations for real analytic unbounded solutions

Answers positively to Burns' question on homogeneous polynomials

Generalizes Wong's classification of weighted circular domains

Abstract

We study the problem of the existence and the holomorphicity of the Monge-Amp\`ere foliation associated to a plurisubharmonic solutions of the complex homogeneous Monge-Amp\`ere 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-Amp\`ere equation and we obtain a generalization the work of P.M. Wong on the classification of complete weighted circular domains.