  A systematic investigation is presented of grain boundaries and grain
boundary networks in two dimensional flexible membranes with crystalline order.
An isolated grain boundary undergoes a buckling transition at a critical value
of the elastic constants, but, contrary to previous findings, the shape of the
buckled membrane is shown to be asymptotically flat. This is in general not
true in the case of a network of intersecting grain boundaries, where each
intersection behaves as the source of a long range stress field. Unlike the
case of isolated dislocations or disclinations, the energy associated with
these stresses is finite; they can, however, destabilize the flat phase. The
buckled phase can be modeled by an Ising spin-glass with long range
antiferromagnetic interactions. These findings may be relevant to the
understanding of the wrinkling transition in partially polymerized vesicles
reported in recent experiments.
