Surfaces in the 3-dimensional Lorentz-Minkowski space satisfying \(\Delta x = Ax + B\) (Q1204315)

The authors give a classification of surfaces of the 3-dimensional Minkowski space \(L^ 3\) satisfying \(\Delta x = Ax + B\) where \(\Delta\) is the Laplacian on the surface, \(x\) represents the isometric immersion, \(A\) is an endomorphism of \(L^ 3\) and \(B\) is a constant vector. The main result is the following: Let \(x: M^ 2_ s \to L^ 3\) be an isometric immersion. Then \(\Delta x = Ax + B\) if and only if one of the following statements holds true: (1) \(M^ 2_ s\) has zero mean curvature everywhere. (2) \(M^ 2_ s\) is an open piece of one of the following surfaces \(L \times S^ 1(r)\), \(H^ 1(r) \times R\), \(S^ 1_ 1(r) \times R\), \(H^ 2(r)\), \(S^ 2_ 1(r)\). Similar results have been obtained by Dillen, Pas, Verstraelen for surfaces in \(E^ 3\).