Minimally immersed Legendrian surfaces in Sasakian 5-manifolds (Q5929128)

Let \(M(\eta,\xi,\varphi,g)\) be a Sasakian 5-manifold with contact form \(\eta\), Reeb vector field \(\xi\) and Sasakian metric \(g\). Minimal Legendrian surfaces of \(M\) are considered. For a Legendrian surface \(\Sigma\) of \(M\), introduce isothermal coordinates \(x,y\) so that \(\Sigma\) is a Riemann surface with complex coordinate \(z=x+iy\). Define the cubic differential \(Q(z)=\left\langle\sigma\left({\partial\over\partial z},{\partial\over\partial z}\right),\varphi\left({\partial\over\partial z}\right)\right\rangle dz^3\), where \(\sigma\) is the second fundamental form. The differential \(Q(z)\) is holomorphic under the following curvature hypothesis \(g(\overline R({\partial \over \partial \overline z},{\partial \over\partial z}){\partial \over\partial z}, \varphi ({\partial\over \partial z}))=0\), where \(\overline R\) is the Riemann curvature tensor of \((M,g)\). The main results are as follows: Let \(X\) be a compact, connected and orientable Legendrian surface minimally immersed in a Sasakian 5-manifold \(M\), which satisfies the above curvature condition. If \(\Sigma\) has genus zero, then it is totally geodesic. If the Gaussian curvature \(K\geq 0\), then either \(\Sigma\) is totally geodesic or \((\Sigma,g)\) is flat and the second fundamental form \(\sigma\) has constant norm. If \(K\leq 0\) and \(M\) has positive sectional curvature, then \((\Sigma,g)\) is a flat torus. In the case where \(\Sigma\) is complete and open, some similar results are shown. These results generalize those of \textit{S. Yamaguchi}, \textit{M. Kon} and \textit{Y. Miyahara} [Proc. Am. Math. Soc. 54, 276-280 (1976; Zbl 0283.53044)] where the ambient space \(M\) is assumed to be a Sasakian space form of dimension five.