Geometrisation of 3-Manifolds (EMS Tracts in Mathematics) by Laurent Bessieres, Gerard Besson, Michel Boileau

By Laurent Bessieres, Gerard Besson, Michel Boileau

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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. 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.

R; ı; Ä/bubbling-off on Œ0; T  with such initial data. We reduce the proof of this result to three independent propositions, called A, B and C, whose demonstrations occupy all Part II of the book. 4 a long-time existence theorem, obtained more or less by iteration of the first one. 1 Let the constants be fixed Recall that "0 has been fixed in Chapter 3 (cf. 1). 1. 5. 4. 2. Let r > 0. "0 ; C0 /-canonical neighbourhood. 3. Let Ä > 0. t/g is Ä-noncollapsed on all scales less than or equal to 1. 4.

M; g. ˇ"/ 1 ; jbj/ is unscathed and satisfies jRmj 6 2Kst . x; 0/ is the centre of a strong "-neck. Proof. We argue by contradiction, assuming that there exists a number ", a sequence ˇk ! xk ; 0/ is not the centre of a strong "-neck. x; b/ From assumptions (i), (ii) we first deduce that jRmj 6 2 on Mk Œbk Qk 1 ; bk  for all k. ˇk "/ 1 /. ˇk "/ 1 ; bk Qk 1 /. xk ; 0/. xk ; bk / D 1 implies that Qk ! 1. M1 ; g1 . b 1; 0, where b ´ lim bk 2 Œ 3=4; 0. b 1=2/; x1 /. S 2 R; gcyl . 1=2/; /. b 1=2/; x1 / is isometric to this cylinder.

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