  I study Gromov-Hausdorff limits of complex curves endowed with singular flat
metrics of constant diameter. I formulate a criterion that the limit is
collapsed in terms of a certain piecewise affine weight function on the dual
intersection complex of a semi-stable model of the degeneration introduced by
Kontsevich and Soibelman. I describe the collapsed and non-collapsed limits,
which are, respectively, metric graphs and finite collections of complex curves
with flat metrics glued along finitely many points. I show that the collapsed
limit of any positive genus can occur.
