1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

NO FINITE TIME LOCAL COLLAPSING 257 PROOF. (i) ===?(ii). We prove (ii) by contradiction. Suppose g (t) is locally collapsing a ...

258 6. ENTROPY AND NO LOCAL COLLAPSING we have injg(t) (p) 2: er. Corollary 6.62, i.e., Perelman's no local collapsing theorem, ...

NO FINITE TIME LOCAL COLLAPSING 259 Riemannian manifold, p EM and r E (0, p] are such that Re 2': -c 1 (n) r-^2 and RS c1 (n) ...

260 6. ENTROPY AND NO LOCAL COLLAPSING where d 9 ( x, p) is the distance function and the constant c = c ( n, g, x, r) is chosen ...

NO FINITE TIME LOCAL COLLAPSING 261 since r^2 R::::; ci(n) in B(p,r), supp(w) c B(p,r), and JMw^2 dμ = 1. By (6.86), we have a ...

262 6. ENTROPY AND NO LOCAL COLLAPSING The second part follows from that fact that if for some K, > 0 and r > 0 the metric ...

NO FINITE TIME LOCAL COLLAPSING 263 k. We can apply Remark 6.66 to the ball B (p, r /2k) and get Vol B (p, r /2k) 2: "'o (r /2 ...

264 6. ENTROPY AND NO LOCAL COLLAPSING JKiT for all t E [-Kiti, Ki (T - ti)). Since limi-+oo JKiT = oo, goo (t) is A;-noncollaps ...

IMPROVED VERSION OF NLC AND DIAMETER CONTROL 265 p EM, and r > 0, we have ( 6. 96 ) μ (g, r2) <log VolB (p, r) + 36 + lB ...

266 6. ENTROPY AND NO LOCAL COLLAPSING the quantity MR(P, r) is a well-defined finite number for 0 < r < oo. Clearly, if r ...

IMPROVED VERSION OF NLC AND DIAMETER CONTROL 267 Hence VolB(p,s) > (~)nVolB(p,s/2) sn - 2 (s/2t > (~)nk VolB (p,s/2k) - ...

268 6. ENTROPY AND NO LOCAL COLLAPSING THEOREM 6. 75 (Topping). Let n 2: 3 and let (ktn, 9) be a closed Rie- mannian manifold wi ...

IMPROVED VERSION OF NLC AND DIAMETER CONTROL 269 The theorem follows from plugging in the definition of 8 (g). D Now we can ap ...

270 6. ENTROPY AND NO LOCAL COLLAPSING μ(9, r^2 ) S JM r^2 (4l'Vwl^2 + Rw^2 ) dμ JM (log ( w^2 ) + i log( 47rr^2 ) + n) w^2 dμ ...

IMPROVED VERSION OF NLC AND DIAMETER CONTROL 271 LEMMA 6.80. Let u be a positive solution to the heat equation (:t-.6.)u=O on ...

272 6. ENTROPY AND NO LOCAL COLLAPSING PROPOSITION 6.81 (Weaker version using the heat equation). If (.Mn, g) is a complete nonc ...

SOME FURTHER CALCULATIONS RELATED TO :F AND W 273 Applying the Holder inequality, JM IY'fl 4 udμ 2 (JM IY'fl^2 udμ) 2 = 16,\i, ...

274 6. ENTROPY AND NO LOCAL COLLAPSING is a bounded nonnegative solution to the Lichnerowicz Laplacian heat equa- tion gtv = t:J ...

SOME FURTHER CALCULATIONS RELATED TO :F AND W 275 LEMMA 6.87. Under the equations gtgij = -2Rij and {) 8t Xi = jj.dXi - Y'iR + ...

276 6. ENTROPY AND NO LOCAL COLLAPSING COROLLARY 6.89. If Zt9ij = -2~j and ~~ =~f+a(R+2~f-1Vf1 2 ), for some a E JR, then the en ...