Pluricomplex Green function

Summary

Let $\Omega \subseteq \C^n$ be a domain and $w \in \Omega$. The pluricomplex Green function $g_\Omega(\cdot, w)$ is the largest negative plurisubharmonic function on $\Omega$ with a logarithmic pole at $w$. It was introduced by Maciej Klimek as the analogue of the classical Green's function of a planar domain. On bounded hyperconvex domains it is continuous and solves a homogeneous complex Monge–Ampère equation off the pole, with a Dirac mass at the pole. On bounded convex domains it is expressed through the Kobayashi distance, by a theorem of Lempert. The version with pole at infinity for a compact set is the Siciak extremal function.