On minimal surfaces with the Ricci condition in space forms (Q1345374)

A surface \((M,ds^ 2)\) is said to satisfy the Ricci condition with respect to \(c\) if its Gaussian curvature \(K\) satisfies \(K \leq c\) and the metric \(d \widehat{s}^ 2 = \sqrt {s-K} ds^ 2\) is flat. If \((M, ds^ 2)\) is simply connected with curvature everywhere different from \(c\), then it can be minimally embedded into a space form \(X^ 3(c)\) if and only if it satisfies the Ricci condition with respect to \(c\). The paper under review is a survey of the research of the author on the problem of classifying those minimal surfaces in \(X^ N (c)\) whose induced metric satsifies the Ricci condition with respect to \(c\), 0[see Tsukuba J. Math. 16, No. 1, 161-167 (1992; Zbl 0769.53036) and Proc. Am. Math. Soc. 121, No. 2, 573-577 (1994; Zbl 0807.53048)]. One result of the author is that a minimal surface in \(X^ 4 (c)\) satisfying the Ricci condition with respect to \(c\) must lie in a totally geodesic \(X^ 3 (c)\).