Submanifolds in a Sasakian manifold \(\mathbb{R}^{2n+1}(-3)\) whose \(\phi\)-mean curvature vectors are eigenvectors (Q1396041)

Let \(\mathbb{R}^{2n+1}(-3)\) be a Sasakian manifold, \(x: M^m\to \mathbb{R}^{2n+1}(-3)\) be an isometric immersion. \textit{C. Baikoussis} and \textit{D. E. Blair} proved that if the position vector field \(x\) satisfies \(\Delta x_\phi= \lambda x_\phi\) for some constant \(\lambda\), then \(x\) is either a minimal submanifold in \(\mathbb{R}^{2n+1}(-3)\) or a minimal submanifold in some hypercylinder of \(\mathbb{R}^{2n+1}(-3)\) [Bull. Inst. Math., Acad. Sin. 19, 327--350 (1991; Zbl 0753.53030)]. In this paper the author studies Legendre submanifolds and contact CR hypersurfaces satisfying \(\Delta H_\phi= \lambda H_\phi\), where \(H_\phi\) is \(\phi\)-mean curvature vector. He gets some uniqueness results under some additional condition. For example, he proved that: Let \(M^{2n}\) be a contact mixed geodesic hypersurface of \(\mathbb{R}^{2n+1}(-3)\) with at most five distinct principal curvatures. If \(M^{2n}\) satisfies \(\Delta H_\phi= 0\), then it is minimal.