  The purpose of this article is to classify the real hypersurfaces in complex
space forms of dimension 2 that are both Levi-flat and minimal. The main
results are as follows:
  When the curvature of the complex space form is nonzero, there is a
1-parameter family of such hypersurfaces. Specifically, for each one-parameter
subgroup of the isometry group of the complex space form, there is an
essentially unique example that is invariant under this one-parameter subgroup.
  On the other hand, when the curvature of the space form is zero, i.e., when
the space form is complex 2-space with its standard flat metric, there is an
additional `exceptional' example that has no continuous symmetries but is
invariant under a lattice of translations. Up to isometry and homothety, this
is the unique example with no continuous symmetries.
