  In the framework of nuclear energy density functional (EDF) methods, many
nuclear phenomena are related to the deformation of intrinsic states. Their
accurate modeling relies on the correct description of the change of nuclear
binding energy with deformation. The two most important contributions to the
deformation energy have their origin in shell effects that are correlated to
the spectrum of single-particle states, and the deformability of nuclear
matter, that can be characterized by a model-dependent surface energy
coefficient a_{surf}.
  With the goal of improving the global performance of nuclear EDFs through
fine-tuning of their deformation properties, the purpose of this study is
threefold. First, to analyze the impact of systematic variations of a_{surf} on
properties of nuclei; second, to identify observables that can be safely used
to narrow down the range of appropriate values of a_{surf} to be targeted in
future parameter fits; third, to analyze the interdependence of a_{surf} with
other properties of a nuclear EDF.
  Results for a large variety of relevant observables of deformed nuclei
obtained from self-consistent mean-field calculations with a set of
purpose-built SLy5sX parameterizations of the Skyrme EDF are correlated with
the value of a_{surf}.
  The three main conclusions are that there is an evident preference for a
comparatively low value of a_{surf}, as expected from the performance of
existing parameterizations; that the isospin dependence of the surface energy
also needs further finetuning in order to describe trends across the chart of
nuclei; and that a satisfying simultaneous description of fission barriers and
superdeformed states requires a better description of the single-particle
spectra.
  [Note: The abstract has been abbreviated because of length restrictions
imposed by the arXiv. See the paper for the full abstract.]
