PomeLemma 5.14.14.741322DD572F Let $\varphi \in \mathrm{PSH}((\Delta ^*)^ n)$ be an $(S^1)^ n$-invariant plurisubharmonic function. Then $\varphi $ is finite everywhere.
Lemma 5.14.14741322DD572F
Proof. It clearly suffices to handle the case $n=1$. In this case, by [Theorem 2.12, HK76], the map
\[ \log r\mapsto \int _{0}^{1} \varphi \left(r\mathrm{e}^{2\pi \mathrm{i}\theta }\right)\, \mathrm{d}\theta =\varphi (r) \]
is a convex function of $\log r$. So, the set $\{ r\in (0,1): \varphi (r)=-\infty \} $ is convex. But $\varphi $ is almost everywhere finite by proposition 5.14.1. Since $\varphi $ is $S^1$-invariant, $\varphi |_{(0,1)}$ is almost everywhere finite. It follows from the convexity that it is everywhere finite. $\square$
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