1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

3. HEAT KERNEL ASYMPTOTICS FOR A TIME-DEPENDENT METRIC 281 REMARK 24.23. Below we shall use the following variant of (24.47). In ...

282 24. HEAT KERNEL FOR EVOLVING METRICS of a C^2 function f. Using giJ = <\j +o (r (x)^2 ), we compute using (23.113) that T ...

HEAT KERNEL ASYMPTOTICS FOR A TIME-DEPENDENT METRIC 283 Recall that by assumption (24.11), we have o'l/Jo OT (y,y,r) = 0. Henc ...

284 24. HEAT KERNEL FOR EVOLVING METRICS and by (24.47) we obtain :T (las Z; (r (s)) ds) = O (s^2 ), it follows from (24.54) tha ...

CHARACTERIZING RICCI FLOW BY ASYMPTOTICS OF HEAT KERNEL 285 LEMMA 24.25 (Asymptotic expansion for '1/J1). '1/J1 (x, y, T) (24. ...

286 24. HEAT KERNEL FOR EVOLVING METRlCS From (24.58) we derive 8f 2 n (24.61) ar -l:lg(T)f +IV' fl - R + 27 = 0. Define J: M x ...

CHARACTERIZING RICCI FLOW BY ASYMPTOTICS OF HEAT KERNEL 287 By Theorem 24.21, for T small and for x and y close, HN is a good ...

288 24. HEAT KERNEL FOR EVOLVING METRICS and respectively. Summing the above two formulas yields the following expres- sion for ...

CHARACTERIZING RICCI FLOW BY ASYMPTOTICS OF HEAT KERNEL 289 Applying (24.72) and (24.73) to (24.71) yields (24.74) W = -n + (R ...

290 24. HEAT KERNEL FOR EVOLVING METRICS 5. Heat kernel on noncompact manifolds Thus far in this chapter and the previous chapte ...

HEAT KERNEL ON NONCOMPACT MANIFOLDS 291 isometric to the product of a metric on 8M with an interval; we may do this in a way w ...

292 24. HEAT KERNEL FOR EVOLVING METRICS THEOREM 24.32 (Existence of Dirichlet heat kernel). The function H defined by (24. 75) ...

HEAT KERNEL ON NONCOMPACT MANIFOLDS 293 Note that (1) compared to formula (24.76) in the interior, we have the extra 'jump' te ...

294 24. HEAT KERNEL FOR EVOLVING METRICS Claim. The series in (24.83) converges uniformly on 8M x [O, T] to a continuous functio ...

HEAT KERNEL ON NONCOMPACT MANIFOLDS 295 for k E N and by (24.87). Indeed, assuming (24.88) holds for some k EN, we have Ak+l ( ...

296 24. HEAT KERNEL FOR EVOLVING METRICS On the other hand, we shall obtain a better estimate for the normal deriv- ative of ii ...

HEAT KERNEL ON NONOOMPAOT MANIFOLDS 297 and l _a -_a I:::; VT-0'. 8vy,T 8vy,CT REMARK 24.37. Also note that by (24.93c) we hav ...

298 24. HEAT KERNEL FOR EVOLVING METRICS where C < oo is independent of k, f3k, /k and where we used d-p'Yk s CJ'k/^2 d-;'Yk ...

HEAT KERNEL ON NONCOMPACT MANIFOLDS 299 Now suppose (24.102) 2a -1 < /k Sn - 2a. Regarding the integral in (24.98b), since ...

300 24. HEAT KERNEL FOR EVOLVING METRICS Since f ( C')£ r£ (1 - a) r (1 - ,6) ( r - aY(l-a)-,6 e=o r (1 + £ (1 - a) - ,6) _ -,6^ ...