  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 which we prove the short-time existence of Ricci flow
solutions on noncompact Riemannian manifolds with at most quadratic curvature
growth, whose 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.
