The authors study minimum sets for singular plurisubharmonic functions. In the planar case they prove the following criterion:
Any Jordan curve that can be parametrized so that \(A|z-s|^\gamma\leq|\varphi(t)-\varphi(s)|\leq B|t-s|^\gamma\) with \(\gamma\leq1\) such that \(\frac{2^{1-\gamma}}A<\frac1{\sin(\frac18(\sqrt{17}-1)\pi)}\), is a minimum set of a strictly subharmonic function on \(\mathbb C\setminus\{a\}\) for some \(a\) in the bounded component of the complement of the curve.
Special stress is put on the case of Koch curves.
In the second part of the paper the authors discuss the role played by the so-called “differential tests”. Let \(\varphi\) be an upper semicontinuous function on a domain \(\Omega\subset\mathbb C^n\). We say that \(\varphi\) allows a differential test from above at a point \(z\in\Omega\) if there exists a \(\mathcal C^2\) function \(q\) on a neighborhood \(V\subset\Omega\) of \(z\) such that \(\varphi(z)-q(z)=\sup_{w\in V}(\varphi(w)-q(w))\). The authors prove the following.
– There are sets \(E\subset\Omega\) of Hausdorff dimension \(>2n-1\) and functions \(\varphi\in\mathcal{PSH}(\Omega)\) such \(\varphi\) does not admit a differential test from above at any point of \(E\).
– If \(\varphi\in\mathcal{PSH}(\Omega)\), then \(\varphi\) allows a differential test from above at almost all points of \(\Omega\).
Moreover, the authors present a method of producing differential tests for certain plurisubharmonic functions.