Related papers: Rigidity theorem by the minimal point of the Bergm…

For any bounded convex domain $\Omega$ with $C^{2}$ boundary in $\mathbb{C}^{n}$, we show that there exist positive constants $C_{1}$ and $C_{2}$ such that \[ C_{1}\sqrt{\dfrac{K\left(w,w\right)}{\delta\left(w\right)}}\leq\left\Vert…

Our main result introduces a new way to characterize two-dimensional finite ball quotients by algebraicity of their Bergman kernels. This characterization is particular to dimension two and fails in higher dimensions, as is illustrated by a…

We show that the $p-$Bergman kernel $K_p(z)$ on a bounded domain $\Omega$ is of locally $C^{1,1}$ for $p\geq1$.The proof is based on the locally Lipschitz continuity of the off-diagonal $p-$Bergman kernel $K_p(\zeta,z)$ for fixed $\zeta\in…

This paper studies Fefferman's program \cite{F3} of expressing the singularity of the Bergman kernel, for smoothly bounded strictly pseudoconvex domains $\Omega\subset\C^n$, in terms of local biholomorphic invariants of the boundary. By…

In recent work by Zimmer it was proved that if $\Omega\subset\mathbb C^n$ is a bounded convex domain with $C^\infty$-smooth boundary, then $\Omega$ is strictly pseudoconvex provided that the squeezing function approaches one as one…

The Fefferman--Szeg\H{o} metric \(g_{\operatorname{FS}}^\Omega\) on a \(C^\infty\)-smooth bounded strongly pseudoconvex domain \(\Omega\subset\mathbb C^n\) is an invariant metric defined via the Fefferman surface measure. For this metric,…

The $p-$Bergman kernel $K_p(\cdot)$ is shown to be of $C^{1,1/2}$ for $1<p<\infty$. An unexpected relation between the off-diagonal $p-$Bergman kernel $K_p(\cdot,z)$ and certain weighted $L^2$ Bergman kernel is given for $1\le p\le 2$. As…

We prove some rigidity theorems for configurations of closed disks. First, fix two collections $\mathcal{C}$ and $\tilde{\mathcal{C}}$ of closed disks in the Riemann sphere $\hat{\mathbb{C}}$, sharing a contact graph which…

In 1983 A. Suslin proved the Quillen-Lichtenbaum conjecture about algebraic K-theory of algebraically closed fields. The proof was based on a theorem called the Suslin rigidity theorem. In the present paper we prove the rigidity theorem for…

Let $\Omega$ be a bounded, convex, centrally symmetric in $\mathbb{R}^{2}$ with a connected $C^{2,\epsilon}$ ($\epsilon\in(0,1)$) boundary. We show that, if the following overdetermined elliptic problem \begin{equation} -\Delta u=\alpha…

Let $\Omega \subset \mathbb{C}^n$ be a domain whose Bergman space contains all holomorphic monomials. We derive sufficient conditions for $\Omega$ to be Reinhardt, complete Reinhardt, circular or Hartogs in terms of the orthogonality…

It is a well-known fact that in a Lipschitz domain \Omega\subset R^n a p-Hardy inequality, with weight d(x,\partial\Omega)^\beta, holds for all u\in C_0^\infty(\Omega) whenever \beta<p-1. We show that actually the same is true under the…

We prove several rigidity results for corona $C^*$-algebras and \v{C}ech-Stone remainders under the assumption of Forcing Axioms. In particular, we prove that a strong version of Todor\v{c}evi\'c's $\OCA$ and Martin's Axiom at level…

We consider the Szeg\"o kernel for non-pseudoconvex domains in C^2 given by \Omega = {(z,w): Im w > b(Re z)} for b a non-convex even-degree polynomial with positive leading coefficient. This is an extension of results previously obtained by…

It is proved that for any domain in $\mathbb C^n$ the Caratheodory--Eisenman volume is comparable with the volume of the indicatrix of the Caratheodory metric up to small/large constants depending only on $n.$ Then the "multidimensional…

Let $\Omega$ be a pseudoconvex domain in $\mathbb C^n$ satisfying an $f$-property for some function $f$. We show that the Bergman metric associated to $\Omega$ has the lower bound $\tilde g(\delta_\Omega(z)^{-1})$ where $\delta_\Omega(z)$…

We prove `polynomial in $k$' bounds on the size of the Bergman kernel for the space of holomorphic Siegel cusp forms of degree $n$ and weight $k$. When $n=1,2$ our bounds agree with the conjectural bounds on the aforementioned size, while…

The Bergman theory of domains $\{ |{z_{1} |^{\gamma}} < |{z_{2}} | < 1 \}$ in $\mathbb{C}^2$ is studied for certain values of $\gamma$, including all positive integers. For such $\gamma$, we obtain a closed form expression for the Bergman…

Let $(R,\mathfrak{m},\mathbb{k})$ be an equicharacteristic one-dimensional complete local domain over an algebraically closed field $\mathbb{k}$ of characteristic 0. R. Berger conjectured that R is regular if and only if the universally…

Let $\Omega\subset \mathbb{C}^n$ for $n\geq 2$ be a bounded pseudoconvex domain with a $C^2$-smooth boundary. We study the compactness of composition operators on the Bergman spaces of smoothly bounded convex domains. We give a partial…