  We develop Teichmuller theoretical methods to construct new minimal surfaces
in $\BE^3$ by adding handles and planar ends to existing minimal surfaces in
$\BE^3$. We exhibit this method on an interesting class of minimal surfaces
which are likely to be embedded, and have a low degree Gau\ss map for their
genus; the (Weierstrass data) period problem for these surfaces is of arbitrary
dimension.
  In particular, we exhibit a two-parameter family of complete minimal surfaces
in the Euclidean three-space $\BE^3$ which generalize the breakthrough minimal
surface of C. Costa; these new surfaces are embedded (at least) outside a
compact set, and are indexed (roughly) by the number of ends they have and
their genus. They have at most eight self-symmetries despite being of
arbitrarily large genus, and are interesting for a number of reasons. Moreover,
our methods also extend to prove that some natural candidate classes of
surfaces cannot be realized as minimal surfaces in $\BE^3$. As a result of both
aspects of this work, we obtain a classification of a family of surfaces as
either realizable or unrealizable as minimal surfaces.
