Surfaces in Non-flat 3-Space Forms Satisfying $$\square \vec {\textbf{H}}=\lambda \vec {\textbf{H}}

In this paper, we locally classify the surfaces immersed into the non-flat (Riemannian or Lorentzian) 3-space forms satisfying the condition $$\Box \vec {\textbf{H}}=\lambda \vec {\textbf{H}}$$ □ H → = λ H → for a real number $$\lambda $$ λ , where $$\vec {\textbf{H}}$$ H → is the mean curvature vector field and $$\Box $$ □ denotes the Cheng–Yau operator of the surface. We obtain the classification result by proving, at a first step, that the mean curvature function must be constant and, in a second step, we complete the classification.