×

Minimal surfaces with constant curvature and Kähler angle in complex space forms. (English) Zbl 0678.53055

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 \(H_n\mathbb{C}\) \((K=c/4\), \(\cos \theta =0\).

MSC:

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
PDFBibTeX XMLCite
Full Text: DOI