On slant surfaces
Slant immersions in complex geometry were defined by the author [Geometry of slant submanifolds, Leuven: Kath. Univ. Leuven, Dept. Math. 123 p. (1990; Zbl 0716.53006)] as a generalisation of both holomorphic immersions and totally real immersions. In the present paper the author considers a slant submanifold \(M\) of a Kählerian \(m\)-manifold \(M^2(4\varepsilon)\) with a constant holomorphic sectional curvature \(4\varepsilon\) and proves the inequality \[ H^2(p) \geq 2K(p) - 2(1+3\cos^2\theta)\varepsilon, \quad p\in M, \] where \(H\) and \(K\) are the mean curvature and the Gauss curvature of \(M\), respectively, and \(\theta\) is the slant angle. Then he proves that proper surfaces in \(\mathbb{C}^2\) are minimal if and only if they are special slant. Finally, he determines proper slant surfaces incomplex space forms satisfying the equality case of the above inequality.
scientific article; zbMATH DE number 1339905