Gromov-Hausdorff limits of flat Riemannian surfaces

Abstract

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.