Ohnita, Yoshihiro Minimal surfaces with constant curvature and Kähler angle in complex space forms. (English) Zbl 0678.53055 Tsukuba J. Math. 13, No. 1, 191-207 (1989). For an isometric immersion \(\phi\) of a Riemann surface \(M\) into a Kähler manifold \(N\), the Kähler angle \(\theta\) of \(\xi\) is defined to be the angle determined by \(Jd\phi(\partial /\partial x)\) and \(d\phi(\partial /\partial y)\), where \(z=x+iy\) is the local complex coordinate on \(M\) and \(J\) denotes the complex structure of \(N\). An attempt is made here to classify minimal surfaces of complex space forms, by setting the conditions that they are of constant Gaussian curvature \(K\) and with constant Kähler angle. The classification is as follows: Suppose \(N\) is \(P_n\mathbb{C}\). Then, (1) if \(K>0\), there exists a certain \(k\), \(0\le k\le n\) such that \(K=c/\{2k(n - k)+n\},\quad \cos \theta =K(n-2k)/c\) and \(M\) is an open submanifold of a minimal 2-sphere with Gaussian curvature \(k\) in \(P_n\mathbb{C}\). (2) If \(K=0\), then \(\cos \theta=0\), hence \(M\) is totally real. (3) \(K<0\) never occurs. Suppose \(N\) is \(H_n\mathbb{C}\). Then \(M\) is an open submanifold of \(H_1\mathbb{C}\) in \(H_n\mathbb{C}\) \((K=c\), \(\cos\theta =1)\), that is, totally geodesic or \(H_2\mathbb{R}\) in \(H_n\mathbb{C}\) \((K=c/4\), \(\cos \theta =0\). Reviewer: Tanjiro Okubo (Victoria) Cited in 2 ReviewsCited in 7 Documents MSC: 53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) Keywords:isometric immersion; minimal surfaces; complex space forms; Kähler angle PDFBibTeX XMLCite \textit{Y. Ohnita}, Tsukuba J. Math. 13, No. 1, 191--207 (1989; Zbl 0678.53055) Full Text: DOI