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13). M; g. t // D 0 for all t . 1). M; g. x; 0/ D 1 is compact for the pointed convergence of flows. 1 and Section 43). Every 2-dimensional Ä-solution is round. 1). There exists a universal constant Äsol > 0 such that any 3-dimensional Ä-solution is round or a Äsol -solution. 2 Canonical neighbourhoods The cylindrical flow is S 2 R together with the product Ricci flow on . 1; 0, where the first factor is round, normalised so that the scalar curvature at time 0 is identically 1. t /. 6. Let " > 0.

R; ı/-bubbling-off on M . Let t1 < t2 be two singular times. C0 ‚/ 1 . Proof. One can suppose that g. t1 ; t2 . t2 / D ‚. x; t / D ‚=2. tC ; t2 . 5) of p. t2 t1 /: . 1 The statements Recall that we are assuming that M is a closed, orientable, irreducible 3-manifold, and that M is not spherical. 3 asserts that for any T > 0 and any metric g0 on M , there exists a Ricci flow with bubbling-off g. 0/ D g0 . 1. r; ı; Ä/-bubbling-off defined on Œ0; T  with initial condition g0 . 3. 1 to three results, called Propositions A, B, C, which are independent of one another.

5) of p. t2 t1 /: . 1 The statements Recall that we are assuming that M is a closed, orientable, irreducible 3-manifold, and that M is not spherical. 3 asserts that for any T > 0 and any metric g0 on M , there exists a Ricci flow with bubbling-off g. 0/ D g0 . 1. r; ı; Ä/-bubbling-off defined on Œ0; T  with initial condition g0 . 3. 1 to three results, called Propositions A, B, C, which are independent of one another. gC / 6 ‚=2. Its proof consists in putting together the cutoff parameters theorem and the metric surgery theorem, as well as some elementary topological arguments which are needed in order to find the collection of cutoff ı-necks.