On the pluricomplex Poisson kernel and the associated complex Monge--Ampère equation

We give a geometric characterization of the pluricomplex Poisson kernel of bounded strongly linearly convex domains in ${\mathbb{C}}^{n+1}$ in terms of their foliation by all complex geodesic discs whose closure contains a fixed boundary point, thus confirming a conjecture posed by Bracci et al. in 2009, even in greater generality. The proof relies crucially on our previous results obtained in a series of two papers. We also revisit the homogeneous complex Monge–Ampère equation associated with the pluricomplex Poisson kernel and show that its solutions are far from unique in general by constructing a family of pairwise nonproportional continuous solutions on the unit ball in ${\mathbb{C}}^{n+1}$. This sharply contrasts with the uniqueness of solutions to the corresponding equation for the pluricomplex Green function.