Maximal Plurisubharmonic Models

An emph{analytic pair} of dimension $n$ and {it center $V$} is a pair $(V,M)$ where $M$ is a complex manifold of (complex) dimension $n$ and $VssetM$ is a closed totally real analytic submanifold of dimension $n$. To an analytic pair $(V,M)$ we associate the class $PSHU[V,M]$ of the functions $u:Mto[0,pi/4[$ which are plurisubharmonic in $M$ and such that $u(p)=0$ for each $pinV$. If $PSHU[V,M]$ admits a maximal function $u$, the triple $(V,M,u)$ is said to be a {it maximal model}. After defining a pseudo-metric $E_{V,M}$ on the center $V$ of an analytic pair $(V,M)$, whose geometric properties are studied in ref{section:LaMetrica}, we prove (see Theorem ref{thm::MAMaximal}, Theorem ref {LI}) that maximal models provide a natural generalization of the Monge-Amp`ere models introduced by Lempert and Sz"oke in cite{article:LempertSzoke}.