The Compactness Theorem for Riemannian Manifolds

Abstract

The compactness theorem for the Ricci flow tells us that any sequence of complete solutions to the Ricci flow, having uniformly bounded curvature and injectivity radii uniformly bounded from below, contains a convergent subsequence. This result has its roots in the convergence theory developed by Cheeger and Gromov. In many contexts where the latter theory is applied, the regularity is a crucial issue. By contrast, the proof of the compactness theorem for the Ricci flow is greatly aided by the fact that a sequence of solutions to the Ricci flow enjoy excellent regularity properties (which were discussed in the previous chapter). Indeed, it is precisely because bounds on the curvature of a solution to the Ricci flow imply bounds on all derivatives of the curvature that the compactness theorem produces C8-convergence on compact sets.

The compactness result has natural applications in the analysis of singularities of the Ricci flow by ‘blow-up’, discussed here in Sect. 9.5: The idea is to consider shorter and shorter time intervals leading up to a singularity of the Ricci flow, and to rescale the solution on each of these time intervals to obtain solutions on long time intervals with uniformly bounded curvature. The limiting solution obtained from these gives information about the structure of the singularity.

As a remark concerning notation in this chapter, quantities depending on the metric gk or gk(t) will have a subscript k. For instance ?k and Rk denote the Riemannian connection and Riemannian curvature tensor of gk. Quantities without a subscript will depend on the background metric g.