Exceptional minimal surfaces with the Ricci condition
Let \(X^ N(c)\) denote the \(N\)-dimensional simply connected space form of constant curvature \(c\). The present author studies exceptional minimal surfaces in \(X^ N(c)\), which are defined by \textit{G. D. Johnson} in [Pac. J. Math. 149, No. 1, 113-125 (1991; Zbl 0729.53053)]. The main result is as follows: Let \(M\) be an exceptional minimal surface lying fully in \(X^ N(c)\). We denote by \(K\) the Gauss curvature of \(M\) with respect to the induced metric \(ds^ 2\). Suppose that the metric \(\widetilde{ds}^ 2 = \sqrt{c-K}ds^ 2\) is flat at points where \(K < c\). If \(c>0\), then either (i) \(N = 4m + 1\) and \(M\) is flat, or (ii) \(N = 4m + 3\). If \(c < 0\), then \(N = 3\).
scientific article; zbMATH DE number 109334