We suggest that the vortices arising in a Kosterlitz-Thouless phase transition in Liouville theory correspond to transitions between different genera, producing the "plumber's nightmare" and other phases that have been predicted in fluid membrane theory from energetic considerations. This transition has previously been invoked by Cates to explain the degeneration of numerical simulations of Gaussian random surfaces into branched polymers. The difficulty in quantizing Liouville theory for d>1 is conjectured to be due to our insistence on working at a fixed genus.