Minimal 2-tori in \(S^ 4\) whose Gauss images have constant curvature
Let \(M\) be a minimal surface immersed in the \(n\)-dimensional sphere \(S^ n\) of curvature 1 and let \(K\) and \(\widetilde {K}\) denote the Gaussian curvatures of the induced metrics by the immersion and the Obata's Gauss map respectively. In this paper it is given a characterization of the Clifford torus by the Gauss map, among minimal 2-tori in \(S^ 4\). Namely, it is proved the following: Let \(M\) be a minimal 2-torus in \(S^ 4\) with constant \(\widetilde{K}\) except, possibly, at isolated points where \(K = 1\). Then \(M\) is flat and the Clifford torus in a totally geodesic \(S^ 3\).
scientific article; zbMATH DE number 590078
scientific article