Minimal tori in the five-dimensional sphere in \(\mathbb{C}^3\) (Q1175838)

The author finds derivation formulas and Gauss-Codazzi-Ricci equations for a class of real two-dimensional surfaces in a Euclidean space \(E^6\) given by imbeddings \(r: T\to M\subset\mathbb{C}^3\) where \(M\) is a real submanifold of codimension one in \(\mathbb{C}^3\). In the case that the manifold \(M\) is a sphere \(S_R\) of radius \(R\), the author obtains the equation \(u_{z\bar z}=e^{-2u}-e^u\) describing minimal surfaces with the metric \(g=2R^ 2e^ udzd\bar z\). The author also obtains a construction of solutions of this equation and describes the corresponding immersions of two-dimensional tori in the sphere \(S_R\subset\mathbb{C}^3\).