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.