Alan Beardon (Cambridge)

Caratheodory's Kernel Theorem

In 1912 Caratheodory introduced the notion of the `kernel' of a sequence of domains in order to describe the local uniform convergence of a sequence of analytic functions. Very roughly speaking, a sequence $f_n$, defined on $D_n$, converges to $f$, defined on $D$, if $D$ is the kernel of the sequence $(D_n)$, and $D_n \to D$. We shall describe how, and why, it is better to regard this result as a theorem about the convergence of conformal metrics, not functions, and discuss the dominant role played by the hyperbolic metric in these ideas.