1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

UPPER BOUND FOR THE LOCAL ENTROPY f B v dμ 201 3.3. Proof of part (1) of Lemma 22.13. We localize the entropy monotonicity for ...

(^202) 22. TOOLS USED IN PROOF OF PSEUDOLOCALITY by (22.67). Since e-x ::;::: 1 - x, this implies (22.69) JM v h dμg(O) ::S -(31 ...

UPPER BOUND FOR THE LOCAL ENTROPY f B v dμ, 203 Now provided A is chosen so that A 2: 1 + lO(n - l)Q (t -f), then (22.74) impl ...

(^204) 22. TOOLS USED IN PROOF OF PSEUDOLOCALITY Regarding the f1h term above, we have (using \7 H = -H \7 J) r f1h ~ r JM hH dμ ...

UPPER BOUND FOR THE LOCAL ENTROPY fsvdμ 205 Using the definition (22.44) of hand the inequality (<// (s))^2 :=:; 10¢ (s), w ...

206 22. TOOLS USED IN PROOF OF PSEUDOLOCALITY where the first equality is by integration by parts. Hence J r H h dμg(t) I 2: e - ...

LOG SOBOLEV INEQUALITY VIA ISOPERIMETRIC INEQUALITY 207 In this section we prove a sharp form of the logarithmic Sobolev inequ ...

208 22. TOOLS USED IN PROOF OF PSEUDOLOCALITY w^1 ,^2 function by a c^1 function 1/J, then replace 1/J by 11/JI since their weak ...

LOG SOBOLEV INEQUALITY VIA ISOPERIMETRIC INEQUALITY 209 to Mt with H = 1~~1-l and f =~'we have F (t) = Vol 0 (Mt) = f 00 ( { 1 ...

210 22. TOOLS USED IN PROOF OF PSEUDOLOCALITY Assuming this claim and using JM 'lj;^2 djl = fJRn h^2 dμJRn, we have 2:: o, where ...

NOTES AND COMMENTARY 211 The claim now ensues from the following consequences of the co-area for- mula: This also completes th ...

212 22. TOOLS USED IN PROOF OF PSEUDOLOCALITY COROLLARY 22.21 (Isometry group is preserved under Ricci fl.ow). If (Mn,g(t)), t E ...

NOTES AND COMMENTARY Space through a microscope with higher resolution, where Space is now described not by some (Riemannian o ...

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Chapter 23. Heat Kernel for Static Metrics Got a good reason for taking the easy way out now. From "Day Tripper" by The Beatles ...

(^216) 23. HEAT KERNEL FOR STATIC METRICS We say that a fundamental solution H to the heat equation is the min- imal positive fu ...

CONSTRUCTION OF THE PARAMETRIX FOR THE HEAT KERNEL 217 (:IRS, c JR^2 is given by (23.1)) defined by (^23. 7 ) E ( t ) , ( 4 (t ...

(^218) 23. HEAT KERNEL FOR STATIC METRICS Fix y E M. We shall show the existence of the functions { ¢k}f=o in Lemma 23.7 with (2 ...

CONSTRUCTION OF THE PARAMETRJX FOR THE HEAT KERNEL 219 where 9ke ~ g (a/axk,a/axe). Therefore a is C^00 on Minj(g)· Moreover, ...

220 23. HEAT KERNEL FOR STATIC METRICS Nate that for r < inj (g) we have that a is bounded and that (23.19) 8loga 8r -- 8r^8 ...