Operator \(\Delta\)-aK on surfaces (Q1107855)

\textit{D. Fischer-Colbrie} and \textit{R. Schoen} [Commun. Pure Appl. Math. 33, 199-211 (1980; Zbl 0439.53060)] proved that for every 2-dimensional complete Riemannian space (M,g) with Gauß curvature K there exists a(g)\(\in {\mathbb{R}}\), \(0\leq a(g)<1\), such that for \(a\leq a_ 0\) the operator \(\Delta\)-aK has a positive solution, and for \(a>a_ 0\) there is no positive solution. The author proves that \(a_ 0\leq 1/4\) for metrics with \(K\leq 0\). The result is applied to stable degenerate minimal surfaces in \({\mathbb{R}}^ 4\).