Norman Levenberg
The maximum modulus principle and projective hulls
Let $\Omega$ be an open subset of the unit disk $\Delta=\{z\in\mathbb C:|z|<1\}$ and let
$T=\{z\in\mathbb C:|z|=1\}$ denote the unit circle. Let $\mathcal M$ be a vector space of complex-valued, continuous functions on $T\cup\Omega$ containing the constants and closed under multiplication by $z$. Hence $\mathcal M$ contains all holomorphic polynomials. If the functions in $\mathcal M$ satisfy a version of the maximum modulus principle, then one should be able to deduce analyticity of these functions in $\Omega$. For example, one may consider a weak version of this principle:
$$
\text{For all }z_0\in\Omega\text{ there exists }C_{z_0}\text{ such that }
|f(z_0)|\leq C_{z_0}\|f\|_T\text{ for all }f\in\mathcal M.
$$
Suppose $\Omega=\Delta$. When can one conclude that each $f\in\mathcal M$ is holomorphic in $\Delta$? Suppose $\Omega=\Delta\setminus\{0\}$. When can one conclude that each $f\in\mathcal M$ is meromorphic in $\Delta$? In this latter setting we get into issues involving projective hulls of sets in $\mathbb C^2$. We discuss some recent results and conjectures of John Wermer - see his preprint Rudin's Theorem and Projective Hulls on the arXiv - as well as our contributions and dreams.
This is joint work with John Anderson, Joe Cima and Tom Ransford.