On slant surfaces with constant mean curvature in \(\mathbb{C}^2\) (Q1365043)

Let \(M\) be a surface in an almost Hermitian manifold. Then \(M\) is called slant if for every tangent vector \(X\), the angle between \(JX\) and \(T_pM\) is independent of \(X\) and \(p\). Slant surfaces are often also called surfaces with constant Kähler angle. Special cases of slant surfaces include complex curves (\(J\) preserves the tangent space) and totally real surfaces (\(J\) maps the tangent into the normal space). The author obtains the following two theorems. Theorem. A flat proper slant surface with nonzero constant mean curvature in \(\mathbb{C}^2\) is an open portion of a helical cylinder. Theorem. There do not exist proper slant surfaces with nonzero constant mean curvature and nonzero constant Gaussian curvature in \(\mathbb{C}^2\). Remark: \textit{B. Y. Chen} and \textit{L. Vrancken} [Result. Math. 31, 28-39 (1997; Zbl 0871.53016)] show that there exist many slant surfaces with constant mean curvature or constant Gauss curvature.