Gauss curvature of Gaussian image of minimal surfaces
Let \(M\) be a minimal surface lying linearly full in a real space form \(R^ n(c)\) of constant curvature \(c\). The Gauss map of \(M\) degenerates at points where the Gauss curvature \(K\) of \(M\) equals \(c\). So assume \(c- K\neq 0\) on \(M\), and denote the Gauss curvature of the Gauss map by \(\overline K_ G\). It is known that, if \(n=3\), then \(\overline {K_ G} = 1-c/(c-K)\) and if \(c = 0\) and \(n \geq 4\), then \(\overline {K_ G} \leq 2\) and \(\overline {K_ G} = 2\) if and only if \(M\) is a complex curve in \(\mathbb{C}^ 2\). In this paper, the author extends the latter result. In particular, he proves that \(\overline {K_ G}\leq 2-c/(c-K)\) with equality if and only if \(n = 4\) and \(M\) is superminimal (i.e. the image of the unit circle in the tangent plane by the second fundamental form is a circle in the normal plane).
scientific article; zbMATH DE number 409643
scientific article