On the Maximal Rank Problem for the Complex Homogeneous Monge-Amp\`ere Equation

TL;DR

This paper provides examples of boundary conditions for the complex homogeneous Monge-Ampère equation that lead to solutions with degeneracies, highlighting limitations in the maximal rank property.

Contribution

It introduces explicit examples of boundary data causing degeneracy in solutions, advancing understanding of the equation's boundary behavior.

Findings

Solutions can be completely degenerate on open sets

Boundary data can cause failure of maximal rank

Highlights limitations of regular boundary conditions

Abstract

We give examples of regular boundary data for the Dirichlet problem for the Complex Homogeneous Monge-Amp\`ere Equation over the unit disc, whose solution is completely degenerate on a non-empty open set and thus fails to have maximal rank.