Minimal immersions of Kähler manifolds into complex space forms (Q1092428)

As it is well known, holomorphic isometric immersions of Kähler manifolds into Kähler manifolds are minimal immersions. But minimal isometric immersions of Kähler manifolds are not always holomorphic or anti-holomorphic. In the present paper, the author obtains the following results: Theorem 1. Let \(CH^ m(c)\) be an m-dimensional complex hyperbolic space of constant holomorphic sectional curvature c \((c<0)\), and let M be an n- dimensional Kähler manifold such that dim \(M_ c=n\geq 2\). Then, every minimal isometric immersion of M into \(CH^ m(c)\) is a holomorphic or anti-holomorphic immersion. Theorem 2. Let \(f: M\to R^{2n+2}/D\) be an isometric stable minimal immersion of an n-dimensional compact Kähler manifold into a \((2n+2)\)-dimensional flat torus. Assume that \(| R|^ 2\geq \tau^ 2\) holds on M, where R is the curvature tensor and \(\tau\) is the scalar curvature of M. Then, f is holomorphic with respect to some orthogonal complex structure of \(R^{2n+2}/D\). The last result is a generalization of \textit{M. J. Micallef} [J. Differ. Geom. 19, 57-84 (1984; Zbl 0537.32010)].