Let $u$ be a subharmonic function on some open set $G\subset \mathbb{C}$, that is, $u$ is uppersemicontinuous and $\vartriangle u\geq 0$ in the sense of distribution. Let $P=\{z\in G: u(z)=-\infty\}$. I want to know what can one say about $P$ in general. For instance, would it be discrete? This is the case if $u=\log |f|$ for some holomorphic function $f$.
Thanks!
Best Answer
One can say a lot, but this is a difficult subject, and I would refer you to books about potential theory. However, general polar sets need not be discrete. In particular, any countable set is polar, or more generally, any countable union of polar sets is polar. So if $(z_k)$ is any sequence in the complex plane converging to $z_0=0$, then $P = \{z_k\}$ is a polar set with a non-isolated point. (An explicit subharmonic function which is $-\infty$ on $P$ is $u(z) = \sum_k 2^{-k} \log |z-z_k|$.)
On the other hand, all polar sets in the plane are totally disconnected, so they can not contain a continuum like an arc or something.