The pluricomplex Poisson kernel for convex finite type domains

Abstract

Given a bounded convex domain $D\subset {\mathbb{C}}^n$ of finite D'Angelo type and a boundary point $\hat{I}¾\in \partial D$, we prove that the homogeneous complex Monge-Ampère equation $(d{d}^{c}u{)}^n=0$ possesses a continuous strictly negative solution $\hat{I}{\text{c◯}}_{\hat{I}}¾$ that vanishes on $\partial D\setminus \{\hat{I}¾\}$ and has a simple pole at $\hat{I}¾$. We establish that $\hat{I}{\text{c◯}}_{\hat{I}}¾(z)$ equals (up to sign) the normal derivative at $\hat{I}¾$ of the pluricomplex Green function $G_z$, and its sublevel sets are the horospheres centered at $\hat{I}¾$. Moreover, $\hat{I}{\text{c◯}}_{\hat{I}}¾$ satisfies a Phragmen-Lindelöf type-theorem and provides a reproducing formula for plurisubharmonic functions. Consequently, $\hat{I}{\text{c◯}}_{\hat{I}}¾$ serves as a generalisation of the classical Poisson kernel of the unit disc. Our approach, based on metric methods and scaling techniques, allows our results to be applied to strongly convex domains with $C^2$-smooth boundaries as well. In the course of the proof, we also establish a novel estimate of the Kobayashi distance near boundary points.

30 pages - final version to appear in J. London Math. Soc