1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

HEAT KERNEL FOR AN EVOLVING METRIC 341 where CE (0, oo) is as in (25.84) and where we.used fo:::; 1 and (26.31). By applying ( ...

342 26. BOUNDS FOR THE HEAT KERNEL FOR EVOLVING METRICS Lx,T· By definition, His the minimal positive solution to (26.6)-(26.7), ...

HEAT KERNEL FOR AN EVOLVING METRIC 343 PROOF OF LEMMA 26.14. (1) Upper bound. Let {DihEN be an exhaus- tion of M as above and ...

344 26. BOUNDS FOR THE HEAT KERNEL FOR EVOLVING METRICS Integrating this, we find that for any TI < T2 eG1(^72 -v) JM ¢(x)H(x ...

BOUNDS OF THE HEAT KERNEL FOR AN EVOLVING METRIC 345 On the other hand, fix a compact domain/(, c M. We have l H(x,r;z,p)H(z,p ...

346 26. BOUNDS FOR THE HEAT KERNEL FOR EVOLVING METRICS REMARK 26.18. (1) Roughly speaking, estimate (26.42) is sharper for x ne ...

BOUNDS OF THE HEAT KERNEL FOR AN EVOLVING METRIC 347 since T - v ST. Thus c H(x,T;y,v)s "{T 1 B ( VT=v )' vo 9 9 x,- 2 - where ...

348 26. BOUNDS FOR THE HEAT KERNEL FOR EVOLVING METRICS LEMMA 26.21 (Exponential quadratically weighted L^2 -estimate in terms o ...

BOUNDS OF THE HEAT KERNEL FOR AN EVOLVING METRIC 349 Given R > 0, let NR (K) ~ { x EM : dg(O) (x, K) :'.SR} denote the R-ne ...

350 26. BOUNDS FOR THE HEAT KERNEL FOR EVOLVING METRICS Now fix 0 < TI < T2 < T and 0 < RI < R2 such that NR 2 (K ...

BOUNDS OF THE HEAT KERNEL FOR AN· EVOLVING METRIC 351 where"( is as above with f being ("(, A)-regular, and define. Ri ~ ( ~ + ...

352 26. BOUNDS FOR THE HEAT KERNEL FOR EVOLVING METRICS Case 1. log (A j~~~D -fa~; :::::; -log 2. Then (26.59) implies ----BR^2 ...

BOUNDS OF THE HEAT KERNEL FOR AN EVOLVING METRIC 353 by (26.52). Second, we compute r 2 d~(O) (x,/C) }/Ci v (x, r) e Dr dμg(T) ...

354 26. BOUNDS FOR THE HEAT KERNEL FOR EVOLVING METRICS The following says that, in an integral sense, the heat kernel decays ex ...

BOUNDS OF THE HEAT KERNEL FOR AN EVOLVING METRIC 355 Let Vol _ _K_ n-1 B (r) denote the volume of a ball of radius r in the si ...

356 26. BOUNDS FOR THE HEAT KERNEL FOR EVOLVING METRICS and there exists a constant C4 < oo depending only on T and sup IRij ...

BOUNDS OF THE HEAT KERNEL FOR AN EVOLVING METRIC 357 PROOF. We only prove the inequality H(x,T;y,v)::; ( ~) Vol_g B_g x, y ~- ...

358 26. BOUNDS FOR THE HEAT KERNEL FOR EVOLVING METRICS where C4 depends only on n, K, and T. Thus (26.67) implies C3Cf exp ( - ...

BOUNDS OF THE HEAT KERNEL FOR AN EVOLVING METRIC 359 Integrating by parts and throwing away a negative boundary term, we have ...

360 26. BOUNDS FOR THE HEAT KERNEL FOR EVOLVING METRICS PROOF. Let u (x, r) ~ H (x, r; y, v). By the Li-Yau inequality (25.58) w ...