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