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n the classical description of surface diffusion, transport on a curved interface is associated with the Laplace–Beltrami operator acting on a chemical potential (difference). An early attempt to model surface diffusion via the phase-field approach goes back to Cahn, Elliott and Novick-Cohen; they use a scalar mobility approaching zero in the bulk. Similar models have been proposed first on the basis of heuristic ideas and then underpinned by asymptotic analysis. As it turns out, most of these analyses suffer from a subtle flaw, due not to a miscalculation but rather to early termination of the calculation. The asymptotic analysis provides all the equations desired for the correct sharp-interface limit. Unfortunately, it provides an additional equation, which is one restriction too many. Consequences for dynamical simulations of this kind of model are explored numerically. We construct two models based on the introduction of a tensorial mobility that approximate known sharp-interface equations without adding undesired constraints. Numerical simulations suggest superior performance of the new models in at least some situations.