Translation surfaces in the 3-dimensional space satisfying Δ III r i = μ i r i

Translation surfaces of coordinate finite type

“…Then the Beltrami operator with respect to the third fundamental form, after a lengthy computation, can be expressed as follows [2]…”

Rotational hypersurfaces in Lorentz-Minkowski 4-space

“…For instance, finite type surfaces of revolution in a Euclidean 3-space have been classified in [6] and some properties about surfaces of revolution in four dimensions have been given in [17]. In [5], the authors have studied the translation surfaces in the 3-dimensional Euclidean and Lorentz-Minkowski spaces under the condition ∆ III r i = µ i r i , µ i ∈ R, where ∆ III denotes the Laplacian of the surface with respect to the nondegenerate third fundamental form III and in [8], the authors have classified the translation surfaces in three dimensional Galilean space G 3 satisfying ∆ II x i = λ i x i , λ i ∈ R, where ∆ II denotes the Laplacian of the surface with respect to the nondegenerate second fundamental form II (throughout this study, we call the operators ∆ II and ∆ III as second Laplace-Beltrami operator and third Laplace-Beltrami operator, respectively). The general rotational surfaces in Minkowski 4-space and the third Laplace-Beltrami operator and the Gauss map of the rotational hypersurface in Euclidean 4-space have been studied in [10] and [14], respectively.…”

Translation surfaces in affine 3-space

“…It was Takahashi who first proved that the submanifolds satisfying ∆x = −λx with identical λ ∈ R are either minimal submanifolds of E m or minimal submanifolds of hyperspheres S m−1 of E m . For more study of finite type of surfaces (submanifolds), we refer ( [4,8,9,13,21]).…”