1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

CHAPTER 17 Entropy, μ-invariant, and Finite Time Singularities I'll tip rny hat to the new constitution. Frorn "Won't Get Foole ...

2 17. ENTROPY, μ,-INVARIANT, AND FINITE TIME SINGULARITIES 1.1. Perelman's entropy and its associated invariants. In this subsec ...

COMPACT FINITE TIME SINGULARITY MODELS ARE SHRINKERS 3 The v-invariant is (17.10) v(g) ~inf {μ(g,T): TE JR+}. 1.1.2. Monotonic ...

4 17. ENTROPY, μ-INVARIANT, AND FINITE TIME SINGULARITIES LEMMA 17.1 (Logarithmic Sobolev inequality). Let (Mn,g) be a closed Ri ...

COMPACT FINITE TIME SINGULARITY MODELS ARE SHRINKERS 5 (ii) Hence 2 Ca,A.g ( 2 ) =aVolA.g ( 2 )-2/n +4ae2Cs(M,A2g) n sC(a,g) i ...

6 17. ENTROPY, μ-INVARIANT, AND FINITE TIME SINGULARITIES PROOF. By the scaling invariance ofμ (see property (iii) on p. 236 of ...

COMPACT FINITE TIME SINGULARITY MODELS ARE SHRINKERS 7 The following is inequality (3) in Lemma 2.1 of [197] (compare with (6. ...

8 17. ENTROPY, μ-INVARIANT, AND FINITE TIME SINGULARITIES On the other hand, so that JM 4 JV'wJ 2 dμ :S: JM (4 JV'wj 2 + (R-Rmin ...

COMPACT FINITE TIME SINGULARITY MODELS ARE SHRINKERS 9 REMARK 17.9. For an elementary upper bound of the volume of a solu- tio ...

10 17. ENTROPY, μ-INVARIANT, AND FINITE TIME SINGULARITIES On the other hand, it is possible for a finite time singular solution ...

COMPACT FINITE TIME SINGULARITY MODELS ARE SHRINKERS 11 We claim that (17.32) A (g 00 (t)) > 0 for all t E (-oo, O]. To pro ...

12 17. ENTROPY, μ-INVARIANT, AND FINITE TIME SINGULARITIES PROOF. STEP 1. v (gi) is bounded above by a negative constant. By (17 ...

COMPACT FINITE TIME SINGULARITY MODELS ARE SHRINKERS 13 PROOF. Given TE [c-^1 ,CJ, let w with JMw^2 dμ 9 = 1 be a minimizer of ...

14 17. ENTROPY, μ-INVARIANT, AND FINITE TIME SINGULARITIES Sesum [169] considers immortal solutions g (t) to the 'Ricci flow wit ...

BEHAVIOR OF μ (g, T) FOR T SMALL 15 One could conceivably have the strange situation of a Type Ila singular solution which for ...

16 17. ENTROPY, μ-INVARIANT, AND FINITE TIME SINGULARITIES We now rule out μ (g, f) = 0, from which the lemma follows. Suppose μ ...

BEHAVIOR OF μ(g,'T) FOR 'T SMALL 17 PROOF. Suppose that (17.47) is not true. Then there exist s > 0 and a sequence { TihEN ...

18 17. ENTROPY, μ,-INVARIANT, AND FINITE TIME SINGULARITIES STEP 1. For a subsequence, the functions wi ~ wi o <I> i : ui ...

BEHAVIOR OF μ (g, T) FOR T SMALL 19 Since gi = if!igi---+ 9JR.n, by (17.116) below and (17.60), we have for any compact domain ...

20 17. ENTROPY, μ-INVARIANT, AND FINITE TIME SINGULARITIES in ci,a (D, gJRn) for all compact domains D c JR.n. Note that now we ...