By exploiting Perelman’s pseudolocality theorem, we prove a new compactness theorem for Ricci flows. By optimising the theory in the two-dimensional case, and invoking the theory of quasiconformal maps, we establish a new existence theorem which generates a Ricci flow starting at an arbitrary incomplete metric, with Gauss curvature bounded above, on an arbitrary surface. The criterion we assert for well-posedness is that the flow should be complete for all positive times; our discussion of uniqueness also invokes pseudolocality.
Cite this article
Peter Topping, Ricci flow compactness via pseudolocality, and flows with incomplete initial metrics. J. Eur. Math. Soc. 12 (2010), no. 6, pp. 1429–1451