Minimal kernels of weakly complete spaces
The main results of the paper are the following two theorems. (a) Let \(X\) be a weakly complete manifold, i.e. a complex manifold \(X\) such that the set \(\mathcal F(X)\) of all continuous plurisubharmonic exhaustion functions is nonempty. For every \(\varphi\in\mathcal F(X)\), let \(Z^1_\varphi\) denote the set of all \(x\in X\) such that \(\varphi\) is not strictly plurisubharmonic near \(x\). Let \(\varphi\in\mathcal F(X)\) be a minimal function, i.e. \(Z^1_\varphi=\bigcap_{\psi\in\mathcal F(X)}Z^1_\psi\). Then the level sets \(\Sigma^1_c=\Sigma^1\cap\{\varphi=c\}\) have the local maximum property, i.e. for every point \(x\in Y:=\Sigma^1_c\) there exists a neighborhood \(U\) such that for every compact set \(K\subset\subset U\) and for every function \(\psi\) plurisubharmonic in a neighborhood of \(K\) we have \(\max_{Y\cap K}\psi=\max_{Y\cap\partial K}\psi\). (b) If \(\dim X=2\) and \(\varphi\) is a \(\mathcal C^2\)-minimal function, then for every regular value \(c\) of \(\varphi\), the set \(\Sigma^1_c\), is a compact space foliated by Riemann surfaces.
scientific article; zbMATH DE number 2091741