1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

SUPPLEMENTARY MATERIAL: ELEMENTARY TOOLS 261 so that if Mis given by (23.138), then dd: 3 1s=O <let M =tr (A) ( tr 2 (A) - ...

262 23. HEAT KERNEL FOR STATIC METRICS PROOF. By the remark, for every s > 0 there exists Re:< oo independent oft E [a,w] ...

NOTES AND COMMENTARY 263 By the fundamental theorem of calculus, we have d r d 1[ dt JM f (x, f) dμ (x) = dt °' cp (t) dt = cp ...

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Chapter 24. Heat Kernel for Evolving Metrics I guess I'll never learn... another page is turned. From "I'll Wait" by Van Halen ...

266 24. HEAT KERNEL FOR EVOLVING METRICS where Q: Jvl x [O, T]-+ ffi. is a 000 function and where A 7 = A 9 ( 7 ) denotes the La ...

1. HEAT KERNEL FOR A TIME-DEPENDENT METRIC 267 THEOREM 24.2 (Existence of the heat kernel on closed (Mn,g(r))). In the setup abo ...

268 24. HEAT KERNEL FOR EVOLVING METRJCS Analogous to (23.18), we have that in LJ 7 E(O,T] Minj(g(T)) x { r} x [O, r) C M x M x ...

1. HEAT KERNEL FOR A TIME-DEPENDENT METRIC 269 (2) If the { 'lfJk}1;= 0 satisfy (24.15)-(24.16), then HN = E L:1:=o 'l/JkTk is a ...

270 24. HEAT KERNEL FOR EVOLVING METRICS have that equation (24.12) is equivalent to the following equation: (24.18) ( 0 = -rT8l ...

2. EXISTENCE OF THE HEAT KERNEL FOR A TIME-DEPENDENT METRIC 271 where the infimum is taken over all minimal geodesics joining y ...

272 24. HEAT KERNEL FOR EVOLVING METRICS where Fk,£ is a C^00 covariant k-tensor on UrE[O,T] Minj(g(r)) x { T} X [O, T]. EXERCIS ...

EXISTENCE OF THE HEAT KERNEL FOR A TIME-DEPENDENT METRIC 273 2.2. The parametrix convolution series. Similarly to the previous ...

274 24. HEAT KERNEL FOR EVOLVING METRICS Now define Fk ~ F Fk-l fork EN (F^1 ~ F); since convolution is associative, we may wri ...

2. EXISTENCE OF THE HEAT KERNEL FOR A TIME-DEPENDENT METRIC 275 for a E (0, 1). Taking a E a, 1), we have I~~~ (x, r; y, v; O")I ...

276 24. HEAT KERNEL FOR EVOLVING METRICS Now (1) and (2) imply that the integral on the RHS of (24.34) is the sum of two integra ...

EXISTENCE OF THE HEAT KERNEL FOR A TIME-DEPENDENT METRIC 277 The second formula is obvious. The first formula is true since by ...

278 24. HEAT KERNEL FOR EVOLVING METRICS for some constants c > 0 and C < oo. Thus, defining . c~ ( CVol (g (T)) r-l ck =; ...

HEAT KERNEL ASYMPTOTICS FOR A TIME-DEPENDENT METRIC 279 THEOREM 24.21 (Heat kernel expansion for evolving metrics). On a close ...

(^280) 24. HEAT KERNEL FOR EVOLVING METRICS LEMMA 24.22 (Expansion for °;;). Let x~ = x~ (x), so that r;. (x) 2:7= 1 ( x~) 2 . T ...