A nullset for normal functions in several variables

Abstract:

Suppose that $\Omega$ is a domain in ${C^n},E \subset \Omega$ is closed in $\Omega$, and $f:\Omega \backslash E \to {C^ * }$ is a meromorphic function. We show that if $f$ is normal and $E$ is an analytic subvariety or, more generally, of locally finite $(2n - 2)$-dimensional Hausdorff measure in $\Omega$ satisfying a certain geometric condition, then $f$ can be extended to a meromorphic function (= holomorphic mapping) ${f^ * }:\Omega \to {C^ * }$. In the case of a subvariety sufficient, but not necessary, for the geometric condition is that the singularities of $E$ are normal crossings. As a digression, we give a new proof for the following result, due to Parreau in the case $n = 1$: if $f$ is in the Nevanlinna class and $E$ is polar (in ${R^{2n}}$), then $f$ has a meromorphic extension ${f^ * }$ to $\Omega$.