Compactness of Riemannian manifolds and flows
Peter M. Topping · Cambridge University Press eBooks · 2006
Having shown in Section 5.3 that the curvature of a Ricci flow must blow up in magnitude at a singularity, we will now work towards a theory of ‘blowing–up’ whereby we rescale a flow more and more as we get closer and closer to a singularity, and hope that if we rescale by enough to keep the curvature under control, then we can pass to a limit of flows to give a Ricci flow which captures some of what is happening at the singularity. Refer back to Section 1.2.3 for a discussion of rescaling. The first step in this direction is to pin down what it means for a sequence of flows, or indeed of manifolds, to converge. We then need some sort of compactness theorem to allow us to pass to a limit. We will not give the rather long proofs of these compactness results, but refer the reader to. Convergence and compactness of manifolds It is reasonable to suggest that a sequence { g i } of Riemannian metrics on a manifold ℳ should converge to a metric g when g i → g as tensors. However, we would like a notion of convergence of Riemannian manifolds which is diffeomorphism invariant: it should not be affected if we modify each metric g i by an i –dependent diffeomorphism. Once we have asked for such invariance, it is necessary to discuss convergence with respect to a point of reference on each manifold.