A note on minimal surfaces in \(\mathbb{R}^n\) whose Gauss images have constant curvature (Q2729686)

Let \(M\) be a minimal surface in the \(n\)-dimensional Euclidean space \(\mathbb{R}^n\) with Gaussian curvature \(K\) with respect to the induced metric \(ds^2\). The Gauss map from \(M\) to the complex quadric \(Q_{n-2}\) in the \((n-1)\)-dimensional complex projective space \(CP^{n-1}\) of constant holomorpic sectional curvature 2 has the metric \(d\widehat s^2=-Kds^2\) of the curvature \(\widehat K\). The author considers Fujiki's question: let \(m\) be a positive integer; which is the lowest dimension \(n_m\) among \(n\) such that there exists a full minimal surface in \(\mathbb{R}^n\) with \(\widehat K={2\over m}\)? The author proves the following results:NEWLINENEWLINENEWLINEThere exists a full minimal surface in \(\mathbb{R}^7\) with \(\widehat K={2\over 5}\); that is \(n_5=6\) or 7;NEWLINENEWLINENEWLINEThere exists a full minimal surface in \(\mathbb{R}^9\) with \(\widehat K={2\over 7}\); that is, \(n_7=8\) or 9.