Pluripotential Theory and Monge–Ampère Foliations

Abstract

A regular, rank one solution u of the complex homogeneous Monge-Ampère equation (∂ ∂ u) n = 0 on a complex manifold is associated with the Monge-Ampère foliation, given by the complex curves along which u is harmonic. Monge-Ampère foliations find many applications in complex geometry and the selection of a good candidate for the associated Monge-Ampère foliation is always the first step in the construction of well behaved solutions of the complex homogeneous Monge-Ampère equation. Here, after reviewing some basic notions on Monge-Ampère foliations, we concentrate on two main topics. We discuss the construction of (complete) modular data for a large family of complex manifolds, which carry regular pluricomplex Green functions. This class of manifolds naturally includes all smoothly bounded, strictly linearly convex domains and all smoothly bounded, strongly pseudoconvex circular domains of C n . We then report on the problem of defining pluricomplex Green functions in the almost complex setting, providing sufficient conditions on almost complex structures, which ensure existence of almost complex Green pluripotentials and equality between the notions of stationary disks and of Kobayashi extremal disks, and allow extensions of known results to the case of non integrable complex structures.

Monge–Ampère exhaustions of almost homogeneous manifolds

“…We now observe that all local properties of exhaustions τ and u on the manifold X \ S, which have been proven in the literature for some specific cases (as, for instance, when X is a manifold of circular type or a Morimoto-Nagano spaces -see e.g. [34,23,25,27]) are valid for any Monge-Ampère space X . In particular, one can directly check that there is always a well defined vector field Z on X \ S that satisfies the condition dd c τ (JZ, JX) = X(τ ) for any vector field X ∈ T (X \ S) .…”