1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

THE FUNCTIONALS μ AND v 237 LEMMA 6.22 (Euler-Lagrange for minimizer). The Euler-Lagrange equa- tion of (6.49) is (6.51) T ( 2 ...

238 6. ENTROPY AND NO LOCAL COLLAPSING Next we prove the existence of a smooth minimizer for (6.49); compare the proof below wit ...

THE FUNCTIONALSμ AND v 239 Hence we have (6.58) μ (g (t), T (t)) :::; W (g (t), f (t), T (t)) :::; W (g (to), f (to), r (to))= ...

240 6. ENTROPY AND NO LOCAL COLLAPSING (3) Improving on the previous part, show that for any times ti, t2 EI with ti ::s; t2, μ€ ...

THE FUNCTIONALSμ AND v Now Hence II^9 klvk - (<I>J;l)*^900 lloe(wk(supp(f)),(w;;^1 )*goo) = llk [^9 klvk] -^900 lloe(sup ...

242 6. ENTROPY AND NO LOCAL COLLAPSING 3. Shrinking breathers are shrinking gradient Ricci solitons Let (Mn,g(t)), t E [O,T), be ...

SHRINKING BREATHERS ARE SHRINKING GRADIENT RICCI SOLITONS 243 for t E [ti, t2]. Thus f (t) is the minimizer for W (g (t), f (t ...

244 6. ENTROPY AND NO LOCAL COLLAPSING Hence, if T 2: 1, we have (6.63) Since >.(g) > 0, we have lim 7 -+oo μ (g, T) = +oo ...

SHRINKING BREATHERS ARE SHRINKING GRADIENT RICCI SOLITONS 245 (2) Furthermore, if A. (g (t)) > 0 and if v(g(t)) is not stri ...

246 6. ENTROPY AND NO LOCAL COLLAPSING v(g(t2)). Now the theorem follows from Corollary 6.34 and Lemma 6.35(1)- (2), which tell ...

LOGARITHMIC SOBOLEV INEQUALITY 247 COROLLARY 6.38 (Log Sobolev inequality, version 2). For any b > 0, there exists a consta ...

248 6. ENTROPY AND NO LOCAL COLLAPSING and \7 f ~ x - 2 "'¢¢. We compute Ln (~ l\7 Jl2 + f - n) (27r)~n/2 e-f dx = 2 Ln ( ~ lxl ...

LOGARITHMIC SOBOLEV INEQUALITY 249 PROOF OF THEOREM 6.39 FROM THE PROPOSITION. Given f such that (6.74) r (27r)-n/^2 e-f dx = ...

250 6. ENTROPY AND NO LOCAL COLLAPSING w h ere N * -- N-2N 2 an d A N -- ( r(N/r(N) 2 ) ) 2/N 7rN(N-^1 2 ). B S y ter l" mg ' s ...

NO FINITE TIME LOCAL COLLAPSING 251 Now Ln£ (log IF (x) I) IF (x) 12 dx = 1n£ t (f (xk) 2 log lfk (xk) I IT! (xi) 2 ) dx1 · · ...

252 6. ENTROPY AND NO LOCAL COLLAPSING Ricci flow with surgery may lead to the resolution of Thurston's geometriza- tion conject ...

NO FINITE TIME LOCAL COLLAPSING 253 PROOF. We leave it as an exercise to trace through the definition of /'i,- noncollapsed an ...

254 6. ENTROPY AND NO LOCAL COLLAPSING The next exercise shows that on small enough scales 9 is K-noncollapsed for some K. EXERC ...

NO FINITE TIME LOCAL COLLAPSING 255 Rauch) comparison theorem (comparing (Br-2§ (x, 1), r-^2 9) with the ball of radius 8 in t ...

256 6. ENTROPY AND NO LOCAL COLLAPSING 5.2. The no local collapsing theorem and its proof. 5.2.l. No local collapsing theorem an ...