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The pluricomplex Poisson kernel for convex finite type domains

Leandro Arosio, Filippo Bracci, Matteo Fiacchi

TL;DR

The paper constructs a pluricomplex Poisson kernel Ωξ for bounded convex domains of finite type, proving the existence of a continuous negative solution Ωξ to the homogeneous Monge–Ampère equation that vanishes on ∂D\{ξ} and blows up at ξ, with Ωξ equal to ( up to sign) the normal derivative of the pluricomplex Green function G_z. It shows that Ωξ has horosphere-level sets, satisfies a generalized Phragmén–Lindelöf principle, and provides a reproducing formula for plurisubharmonic functions, thereby generalizing the classical Poisson kernel of the unit disk. The work establishes a precise Kobayashi-distance asymptotic in terms of Ωξ and links Ωξ to the pluriharmonic measure via a density |Ωξ(z)|^n on ∂D, while also giving a dilation normalization for holomorphic maps between convex finite-type domains. A key methodological novelty is the use of strong asymptoticity of complex geodesics and scaling techniques, which also yields a counterexample showing the Poisson–horofunction formula may fail in non-convex strongly pseudoconvex domains. These results extend the pluripotential toolkit to a broad class of domains and illuminate intrinsic metric–potential relationships in several complex variables.

Abstract

Given a bounded convex domain $D\subset \mathbb C^n$ of finite D'Angelo type and a boundary point $ξ\in \partial D$, we prove that the homogeneous complex Monge-Ampère equation $(dd^cu)^n=0$ possesses a continuous strictly negative solution $Ω_ξ$ that vanishes on $\partial D\setminus \{ξ\}$ and has a simple pole at $ξ$. We establish that $Ω_ξ(z)$ equals (up to sign) the normal derivative at $ξ$ of the pluricomplex Green function $G_z$, and its sublevel sets are the horospheres centered at $ξ$. Moreover, $Ω_ξ$ satisfies a Phragmen-Lindelöf type-theorem and provides a reproducing formula for plurisubharmonic functions. Consequently, $Ω_ξ$ 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.

The pluricomplex Poisson kernel for convex finite type domains

TL;DR

The paper constructs a pluricomplex Poisson kernel Ωξ for bounded convex domains of finite type, proving the existence of a continuous negative solution Ωξ to the homogeneous Monge–Ampère equation that vanishes on ∂D\{ξ} and blows up at ξ, with Ωξ equal to ( up to sign) the normal derivative of the pluricomplex Green function G_z. It shows that Ωξ has horosphere-level sets, satisfies a generalized Phragmén–Lindelöf principle, and provides a reproducing formula for plurisubharmonic functions, thereby generalizing the classical Poisson kernel of the unit disk. The work establishes a precise Kobayashi-distance asymptotic in terms of Ωξ and links Ωξ to the pluriharmonic measure via a density |Ωξ(z)|^n on ∂D, while also giving a dilation normalization for holomorphic maps between convex finite-type domains. A key methodological novelty is the use of strong asymptoticity of complex geodesics and scaling techniques, which also yields a counterexample showing the Poisson–horofunction formula may fail in non-convex strongly pseudoconvex domains. These results extend the pluripotential toolkit to a broad class of domains and illuminate intrinsic metric–potential relationships in several complex variables.

Abstract

Given a bounded convex domain of finite D'Angelo type and a boundary point , we prove that the homogeneous complex Monge-Ampère equation possesses a continuous strictly negative solution that vanishes on and has a simple pole at . We establish that equals (up to sign) the normal derivative at of the pluricomplex Green function , and its sublevel sets are the horospheres centered at . Moreover, satisfies a Phragmen-Lindelöf type-theorem and provides a reproducing formula for plurisubharmonic functions. Consequently, 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 -smooth boundaries as well. In the course of the proof, we also establish a novel estimate of the Kobayashi distance near boundary points.

Paper Structure

This paper contains 18 sections, 34 theorems, 160 equations.

Table of Contents

  1. Introduction
  2. The pluricomplex Green function
  3. Demailly's pluriharmonic measure
  4. The pluricomplex Poisson kernel
  5. Convex domains of finite type
  6. Preliminaries
  7. Consequences of strong asymptoticity
  8. Complex geodesics
  9. Strong asymptoticity
  10. Horofunctions and horospheres
  11. Generalizations of non-tangential limit
  12. Definition of the function $\Omega_\xi$
  13. $-\Omega_\xi$ is the normal derivative of the pluricomplex Green function
  14. $\Omega_\xi$ solves the Monge-Ampère equation
  15. A new estimate for the Kobayashi distance
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $L\geq 2$. Let $D\subset \mathbb{C}^n$ be a bounded convex domain with $C^L$-smooth boundary of finite line type $L$. Then $\Omega_\xi\colon D\to (-\infty,0)$ is a continuous function solving the following Monge--Ampère equation: Moreover,

Theorems & Definitions (69)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1: Locally finite type
  • Definition 2.2: Maximal plurisubharmonic functions
  • Theorem 2.3: Bedford--Taylor BedTay
  • Proposition 2.4
  • Definition 3.1: Complex geodesics
  • Remark 3.2
  • Example 3.3: Complex geodesics of the egg domain
  • Definition 3.4: Strong asymptoticity
  • ...and 59 more