We prove a general existence result for instantaneously complete Ricci flows starting at an arbitrary Riemannian surface which may be incomplete and may have unbounded curvature. We give an explicit formula for the maximal existence time, and describe the asymptotic behaviour in most cases.

We show uniqueness of Ricci flows starting at a surface of uniformly negative curvature, with the assumption that the flows become complete instantaneously. Together with the more general existence result proved in [10], this settles the issue of well-posedness in this class.Comment: 11 page

We show that any noncompact Riemann surface admits a complete Ricci flow g(t), t ∈ [0, ∞), which has unbounded curvature for all t ∈ [0, ∞).