Maximal plurisubharmonic models

TL;DR

This paper introduces maximal plurisubharmonic models, generalizing Monge-Ampere models, by analyzing functions on complex manifolds with totally real submanifolds and establishing related pseudo-metrics.

Contribution

It defines maximal plurisubharmonic models and proves they extend the concept of Monge-Ampere models using a new pseudo-metric on the submanifold.

Findings

Introduction of a pseudo-metric E(V,M) on the center V.

Establishment of maximal functions u in the class associated with (V, M).

Generalization of Monge-Ampere models through maximal plurisubharmonic models.

Abstract

An analytic pair of dimension n and center V is a pair (V, M) where M is a complex manifold of (complex) dimension n and V is a closed totally real analytic submanifold of dimension n. To an analytic pair (V, M) we associate the class of the functions u from M to a positive bounded interval which are plurisubharmonic in M and such that u(p) = 0 for each p in V. If the class admits a maximal function u, the triple (V, M, u) is said to be a maximal plurisubharmonic model. After defining a pseudo-metric E(V,M) on the center V of an analytic pair (V, M) we prove (see Theorem 4.1, Theorem 5.1) that maximal plurisubharmonic models provide a natural generalization of the Monge-Ampere models introduced by Lempert and Szoke in [16].