Pseudohermitian biminimal Legendre surfaces in the 5-dimensional sphere

Abstract

In this paper, we determine nonminimal pseudohermitian biminimal Legendre surfaces in the unit 5-sphere $S^{5}$. In fact, the product of a circle and a helix of order $4$ is realized as a nonminimal pseudohermitian biminimal Legendre immersion into $S^{5}$. In addition, we obtain that there exist no nonminimal pseudohermitian biminimal Legendre surfaces in a 5-dimensional Sasakian space form of non-positive constant holomorphic sectional curvature for the Tanaka--Webster connection.