Precompactness of solutions to the Ricci flow in the absence of injectivity radius estimates

Abstract

Consider a sequence of pointed $n$–dimensional complete Riemannian manifolds $\left\{\left(M_i,g_i\left(t\right),O_i\right)\right\}$ such that $t\in \left[0,T\right]$ are solutions to the Ricci flow and $g_i\left(t\right)$ have uniformly bounded curvatures and derivatives of curvatures. Richard Hamilton showed that if the initial injectivity radii are uniformly bounded below then there is a subsequence which converges to an $n$–dimensional solution to the Ricci flow. We prove a generalization of this theorem where the initial metrics may collapse. Without injectivity radius bounds we must allow for convergence in the Gromov–Hausdorff sense to a space which is not a manifold but only a metric space. We then look at the local geometry of the limit to understand how it relates to the Ricci flow.