Mesures de Monge-Ampère et mesures pluriharmoniques. (Monge-Ampère measures and pluriharmonic measures)
Let \(\Omega\) be a relatively compact open subset in a Stein manifold, and \(n=\dim_{{\mathbb{C}}}\Omega\). Assume that \(\Omega\) is hyperconvex, i.e. that there exists a bounded psh (plurisubharmonic) exhaustion function on \(\Omega\). A ''pluricomplex Green function'' \(u_{\Omega}\) is then naturally defined on \(\Omega \times \Omega:\) For all \(z\in \Omega\), \(u_ z(\zeta):= u_{\Omega}(z,\zeta)\) is the solution of the Dirichlet problem for the complex Monge-Ampère equation \((dd^ cu_ z)^ n=0\) on \(\Omega\) \(\setminus \{z\}\) such that \(u_ z(\zeta)=\log | \zeta - z| +O(1)\) at \(\zeta =z\); \(u_{\Omega}\) is shown to be continuous outside the diagonal and invariant under biholomorphisms. Bedford and Taylor's Monge-Ampère operators are used in conjunction with a general Lelong-Jensen formula previously found by the author [Mem. Soc. Math. Fr., Nouv. Ser. 19, 124 p. (1985; Zbl 0579.32012)] in order to construct an invariant pluricomplex Poisson kernel \(d\mu_ z(\zeta):= (2\pi)^{- n}(dd^ cu_ z(\zeta))^{n-1}\wedge d^ cu_ z(\zeta)|_{\partial \Omega},\) \((z,\zeta)\in \Omega \times \partial \Omega\). Each measure \(\mu_ z\) on \(\partial \Omega\) is such that \(\mu_ z(V)=V(z)\) for every function V pluriharmonic on \(\Omega\) and continuous on \({\bar \Omega}\); furthermore, \(\mu_ z\) is carried by the set of strictly pseudoconvex points of \(\partial \Omega\) if \({\bar \Omega}\) has a \(C^ 2\) psh defining function. The principal part of the singularity of \(d\mu_ z(\zeta)\) on the diagonal of \(\partial \Omega\) is then computed explicitly when \(\Omega\) is strictly pseudoconvex, using an osculation of \(\partial \Omega\) by balls. Through a complexification process, it is finally shown that Monge-Ampère measures provide an explicit formula representing every point of a convex compact subset \(K\subset {\mathbb{R}}^ n\) as a barycenter of the extremal points of K.
scientific article; zbMATH DE number 3957595
scientific article