Helicoidal surfaces in the three-dimensional Lorentz-Minkowski space satisfying \(\Delta ^{II}r_{i} = \lambda _{i}r_{i}\) (Q657004)

Consider a nondegenerate immersion \(r:M^2\rightarrow {\mathbb R}^3_1\) in three dimensional Lorentz-Minkowski space \({\mathbb R}^3_1\). It is proved that there are no helicoidal surfaces fulfilling the property \(\Delta^{\text{II}}r_i = \lambda_i r_i ~(i=1,2,3)\), where \(\lambda_i\in \mathbb R\), \(r_i:M^2\rightarrow {\mathbb R}\) are the coordinate functions of \(r\) and \(\Delta^{\text{II}}\) is the Laplacian with respect to the second fundamental form.