Canonical surgeries in rotationally invariant Ricci flow

Abstract

We construct a rotationally invariant Ricci flow through surgery starting at any closed rotationally invariant Riemannian manifold. We demonstrate that a sequence of such Ricci flows with surgery converges to a Ricci flow spacetime in the sense of [32]. Results of and Haslhofer [29] then guarantee the uniqueness and stability of these spacetimes given initial data. We simplify aspects of this proof in our setting, and show that for rotationally invariant Ricci flows, the closeness of spacetimes can be measured by equivariant comparison maps. Finally we show that the blowup rate of the curvature near a singular time for these Ricci flows is bounded by the inverse of remaining time squared. 8 Construction of the comparison domain 9 Construction of the comparison map 10 Uniqueness of singular Ricci Flows 11 Number of bumps in a Ricci flow spacetime 12 Bounding the blowup rate 13 Appendix: Preparatory results in Section 8 of the Bamler-Kleiner paperProof. It is clear that the manifolds described in Definition 2.1 are all orientable, and that the actions of O(n + 1) on these manifolds are all faithful and proper, with principal isotropy subgroup given by O(n).On the other hand, suppose that the connected Riemannian manifold (M n+1 , g) admits a proper and faithful cohomogeneity one action of the group G = O(n + 1), which acts via isometries, and has H = O(n) as the principal isometry subgroup. Let Mreg denote the set of all points whose isotropy subgroups are (conjugacy equivalent to) H. This is an open subset of M , and M * reg = Mreg/O(n + 1) is a smooth one-dimensional manifold, and comes equipped with a Riemannian metric induced by g. The space M *reg is open and dense in M * = M/O(n + 1), which is therefore a smooth one-dimensional manifold with boundary. We split into cases according to the topological type of M * .Case One: M * = R. In this case, M * reg = M * and since R is contractible, M must appear as the trivial bundle R × G/H = R × S n .Case Two: M * = S 1 . In this case, we again have M * reg = M * . Choose any point p * ∈ M/O(n + 1), and consider M \ π −1 (p * ). Then like in Case One, this manifold appears as R × S n . The original manifold is then found by identifying the copies of S n at their two ends. The are only two O(n + 1)-invariant diffeomorphisms on S n , namely, ±I, so there are only two possibilities: S 1 × S n , or a non-trivial S 1 bundle, but this second example is not oriented.Case Three: M * = [0, ∞). Let K be the isotropy of the single non-principal orbit. It is well known that K/H is a sphere, say, S d , and K acts linearly on R d+1 , and transitively on S d . Since H = O(n), it is clear that K = O(n + 1) = G, so that the singular orbit is just a point. The resulting manifold is R n+1 .Case Four: M * = [0, 1]. This is just a gluing of two of the manifolds from Case Three along a principal orbit; the result is the compact S n+1 .It is convenient to note that g|Mreg can always be expressed as a warped Riemannian product.