arXiv subjects

John Treuer

Publications and source records attributed to John Treuer.

2 recordsLinked to original sources

arXiv2021

Rigidity theorem of the Bergman kernel by analytic capacity

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.

math.CV↗

arXiv2020

Rigidity theorem by the minimal point of the Bergman kernel

We use the Suita conjecture (now a theorem) to prove that for any domain $Ω\subset \mathbb{C}$ its Bergman kernel $K(\cdot, \cdot)$ satisfies $K(z_0, z_0) = \hbox{Volume}(Ω)^{-1}$ for some $z_0 \in Ω$ if and only if $Ω$ is either a disk minus a (possibly empty) closed polar set or $\mathbb{C}$ minus a (possibly empty) closed polar set. When $Ω$ is bounded with $C^{\infty}$-boundary, we provide a simple proof of this using the zero set of the Szegö kernel. Finally, we show that this theorem fails to hold in $\mathbb{C}^n$ for $n > 1$ by constructing a bounded complete Reinhardt domain (with algebraic boundary) which is strongly convex and not biholomorphic to the unit ball $\mathbb{B}^n \subset \mathbb{C}^n$.

math.CV↗