  We extend the notion of what it means for a complete Ricci flow to have a
given initial metric, and consider the resulting well-posedness issues that
arise in the 2D case. On one hand we construct examples of nonuniqueness by
showing that surfaces with cusps can evolve either by keeping the cusps or by
contracting them. On the other hand, by adding a noncollapsedness assumption
for the initial metric, we establish a uniqueness result.
