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

David Glickenstein
2003 Geometry and Topology  
Consider a sequence of pointed n-dimensional complete Riemannian manifolds (M_i,g_i(t), O_i) such that t in [0,T] are solutions to the Ricci flow and g_i(t) 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
more » ... apse. 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.
doi:10.2140/gt.2003.7.487 fatcat:vokzfnmsffal7otttmujbn36da