Existence, lifespan, and transfer rate of Ricci flows on manifolds with small Ricci curvature

Abstract:

We show that in dimension 4 and above, the lifespan of Ricci flows depends on the relative smallness of the Ricci curvature compared to the Riemann curvature on the initial manifold. We can generalize this lifespan estimate to the local Ricci flow, using what we prove as the short-time existence of Ricci flow solutions on complete noncompact Riemannian manifolds with at most quadratic curvature growth, where the Ricci curvature and its first two derivatives are sufficiently small in regions where the Riemann curvature is large. Those Ricci flow solutions may have unbounded curvature. Moreover, our method implies that, under some appropriate assumptions, the spatial transfer rate (the rate at which high curvature regions affect low curvature regions) of the Ricci flow resembles that of the heat equation.