Rigidity theorem of the Bergman kernel by analytic capacity

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.

This paper has been withdrawn by the author. It has been superseded and generalized by arXiv:2111.10973 (merged from arXiv:2011.05273 and arXiv:2101.01358)

References in corpus (2)

• Rigidity theorem by the minimal point of the Bergman kernel
• Rigidity theorems by the logarithmic capacity