Convexity of the Bergman Kernels on Convex Domains

TL;DR

For a convex domain $Ω ⊂ ℂ^n$ and a convex weight $φ$, the paper proves that $z \mapsto \log K_{Ω,φ}(z)$ is convex on $Ω$ (allowing $-\infty$). The main method adapts Berndtsson's subharmonicity framework to convex domains by embedding the problem in a higher-dimensional convex domain and transferring convexity back to $Ω$; in the unweighted case ($φ≡0$) a Brunn–Minkowski type inequality is derived, implying $K_Ω(z)^{-1/(2n)}$ is convex. The authors provide necessary and sufficient conditions for strict convexity, distinguishing when $Ω$ contains a real line. These results extend known 1D and ball-domain phenomena to general convex domains and yield sharp exponent bounds in the associated Brunn–Minkowski inequality.

Abstract

Let $Ω$ be a convex domain in $\mathbb{C}^n$ and $\varphi$ a convex function on $Ω$. We prove that $\log{K_{Ω,\varphi}(z)}$ is a convex function (might be identically $-\infty$) on $Ω$, where $K_{Ω,\varphi}$ is the weighted Bergman kernel. When $\varphi\equiv0$, we prove a Brunn-Minkowski type inequality, which further implies that $K_Ω(z)^{-\frac{1}{2n}}$ is a convex function if $Ω$ is convex. Some necessary and sufficient conditions for strictly convexity are also obtained.

Theorems & Definitions (21)

• Theorem 1.1
• Theorem 1.2
• Example 1
• Corollary 1.3
• Corollary 1.4
• Lemma 2.1
• proof
• Theorem 3.1
• proof
• proof : Proof of Theorem \ref{['th:convex']}