Beredn transform and the Laplace–Beltrami operator

Let $\Omega$ be a domain in $\mathbf C,K(x,\bar y)$ its Bergman kernel, $\Delta$ the Laplace–Beltrami operator on $\Omega$, and $\mathcal B$ the Berezin transform on $\Omega$, i.e., the integral operator with the kernel $|K(x,\bar y)|^2/K(y,\bar y)$. For domains that are complete in the Riemannian metric $K(x,\bar x)^{1/2}|dx|$, it is shown that $\mathcal B$ is a function of $\Delta$ if and only if $\mathcal B$ commutes with $\Delta$ if and only if the above metric has constant curvature if and only if $\Omega$ is simply connected. This supplements the results of Berezin [5] and of Unterberger and Upmeier [19] for the unit disc. We also briefly treat the case of weighted Bergman spaces, and indicate a relationship with quantization on $\Omega$.