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In this paper we classify the factorable surfaces in the threedimensional Euclidean space E 3 and Lorentzian E 3 1 under the condition Δri = λiri, where λi ∈ R and Δ denotes the Laplace operator and we obtain the complete classification for those ones.Mathematics Subject Classification (2010). 99Z99, 00A00.

In this paper, we study helicoidal surfaces without parabolic points in the 3-dimensional Lorentz-Minkowski space under the condition Δ II ri = λiri where Δ II is the Laplace operator with respect to the second fundamental form and λi is a real number. We prove that there are no helicoidal surfaces without parabolic points in the 3-dimensional Lorentz-Minkowski space satisfying that condition.Mathematics Subject Classification (2010). 53A05, 53A07, 53C40.

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Abstract. In this paper we study the helicoidal surfaces in the three-dimensional LorentzMinkowski space under the condition II r = Ar, where A is a real 3 × 3 matrix and II is the Laplace operator with respect to the second fundamental form. IntroductionLet E 3 1 denote the three-dimensional Lorentz-Minkowski space, that is, the real vector space R 3 endowed with the Lorentzian metricwhere (x, y, z) are the canonical coordinates in R 3 . Let r : M → E 3 1 be an isometric immersion of a pseudo-Riemannian surface in E 3 1 . The notion of finite-type submanifolds in Euclidean space or pseudo-Euclidean space was introduced by Chen [5]. In terms of Chen's theory, a surface M is said to be of finite type if its coordinate functions are a finite sum of eigenfunctions of its Laplacian . Chen posed the problem of classifying the finite-type surfaces in the three-dimensional Euclidean space E 3 . Further, the notion of finite type can be extended to any smooth functions on a submanifold of a Euclidean space or a pseudo-Euclidean space.In [7], Kaimakamis and Papantoniou classified the first three types of surfaces of revolution without parabolic points in the three-dimensional Lorentz-Minkowski space, which satisfy the conditionwhere Mat(3, R) is the set of 3 × 3 real matrices. They proved that such surfaces are either minimal or Lorentz hyperbolic cylinders or pseudospheres of real or imaginary radius.In [2], Bekkar and Zoubir classified the surfaces of revolution with non-zero Gaussian curvature K G in the three-dimensional Lorentz-Minkowski space E 3 1 , whose component functions are eigenfunctions of their Laplace operator, i.e.