ABSTRACT
In this chapter, we will prove that the Bergman kernelK(z, a) associated to a bounded n-connected domain Ω with C∞ smooth boundary has exactly n − 1 zeroes in Ω as a function of z whenever a ∈ Ω is near enough to the boundary. The transformation rule (16.1) for the Bergman kernels under biholomorphic mappings shows that the number of zeroes of the Bergman kernel is invariant under such mappings. Hence, in view of Lemma 12.1, we may study the problem on domains Ω that have real analytic boundary.
