Rigidity theorem of the Bergman kernel by analytic capacity

TL;DR

This paper provides an elementary proof of a rigidity theorem for the Bergman kernel on planar domains, showing that only disks minus polar sets satisfy a specific kernel-volume relation, and extends results to real ellipsoids.

Contribution

The paper offers a simpler proof of a known theorem, removing the need for several complex variables, and establishes a new lower bound for the Bergman kernel.

Findings

Only disks minus polar sets satisfy the kernel-volume condition.

The unit ball is the only real ellipsoid satisfying the kernel condition.

A new lower bound for the on-diagonal Bergman kernel is derived.

Abstract

In [7], Dong and I proved that the domains $D \subset \mathbb{C}$ of finite volume whose on-diagonal Bergman kernels $K(\cdot, \cdot)$ satisfy $K(z_0, z_0) = Volume(D)^{-1}$ are disks minus closed polar sets. We utilized the solution of the Suita conjecture, a deep theorem of several complex variables. In this note, I present a significantly more elementary proof of this theorem that does not use several complex variables. As a corollary, a new lower bound for the on-diagonal Bergman kernel is given. Finally, I show that the only real ellipsoid in Webster normal form which satisfies $K(0, 0) = Volume(D)^{-1}$ is the unit ball.