Curvature pinching theorems for minimal surfaces in complex Grassmann manifolds (Q5939531)

Let \(f: M^2\to\mathbb{C}\mathbb{P}^n\) a minimal surface in the complex projective space. We define a Kähler angle [see \textit{S. Chern} and \textit{J. G. Wolfson}, Am. J. Math. 105, 59-83 (1983; Zbl 0521.53050)] which measures the failure of \(f\) to be minimal. For isometric minimal immersions into the complex Grassmannians \(G(m,n)\) of \(m\) complex plane in \(\mathbb{C}^n\), there is an analogue of the Kähler angle. The author gives all basic definitions to introduce this concept, and proves some theorems related to it. For instance: NEWLINENEWLINENEWLINELet \(f: M^2\to G(m,n)\) be an isometric minimal surface into \(G(m,n)\) with Kähler angle \(\alpha\) and Gauss curvature \(K\). Assume that NEWLINE\[NEWLINE K\geq \max(4\cos^2 \tfrac d2, 4 \sin^2 \tfrac d2). NEWLINE\]NEWLINE Then: a) Either \(K=4\) and \(f\) is holomorphic or antiholomorphic, or b) \(K=2\), and \(f\) is totally real.