Existence, Lifespan and Transfer Rate of Ricci Flows on Manifolds with Small Ricci Curvature

TL;DR

This paper investigates how the lifespan and transfer rate of Ricci flows on manifolds are influenced by small Ricci curvature relative to Riemann curvature, establishing short-time existence results for certain noncompact manifolds.

Contribution

It extends lifespan estimates to local Ricci flows on noncompact manifolds with quadratic curvature growth and small Ricci curvature, and analyzes the heat-like transfer rate of curvature effects.

Findings

Lifespan depends on Ricci curvature smallness compared to Riemann curvature.

Short-time existence of Ricci flows on noncompact manifolds with unbounded curvature.

Transfer rate of curvature effects resembles heat equation behavior.

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 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.