On minimal immersions of \(R^ 2\) into \(P^ n(C)\)
In this paper, complete minimal surfaces in a Kähler manifold are studied. The first theorem of this paper proves that if the square length of the second fundamental form of a compact minimal surface, \(M^ 2\), in \(P^ n(C)\) is smaller than or equal to \(2\rho\), where \(4\rho\) denotes the constant holomorphic sectional curvature of \(P^ n(C)\), then \(M^ 2\) is superminimal or it is flat and totally real in \(P^ n(C)\). Superminimal surfaces in \(P^ n(C)\) have been studied extensively by several mathematicians and physicists (Chern, Eells, Wolfson, Wood, et al.). Therefore we intent to determine all isometric totally real, minimal immersions of the Euclidean 2-plane \(R^ 2\) into \(P^ n(C).\) Main results of this paper show that they constitute an (n-2)-parameter family and each of them must be an orbit of an abelian Lie subgroup of \(U(n+1)\). These results are proved by applying Chern and Wolfson's theory for minimal surfaces in Kähler manifolds.
scientific article; zbMATH DE number 3897780