1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

DIFFERENTIATING A CONVOLUTION WITH THE PARAMETRIX 241 In particular, (23.88) I ~~ ( x, z, t _ s) I "5. C (i + rz (x)) (t-s)-n/ ...

242 23. HEAT KERNEL FOR STATIC METRICS so that d~ /i = a~i. By the mean value theorem, we have (23.91) l; JN (ri (h), y, s, t) d ...

3. DIFFERENTIATING A CONVOLUTION WITH THE PARAMETRIX 243 where we assumed that {xi} are geodesic coordinates centered at x. Sinc ...

(^244) 23. HEAT KERNEL FOR STATIC METRICS since t - s is bounded from above. However, from this estimate (contrast with (23.88)) ...

DIFFERENTIATING A CONVOLUTION WITH THE PARAMETRIX 245 so that 8 xi (z) -;::;-rd UXz (x, z) = d ( X,Z )" On the other hand, con ...

246 23. HEAT KERNEL FOR STATIC METRICS and \i'\i'd (ei, ej) = \i'\i'd (ei - (ei, V'd) \i'd, ej - (ej, V'd) \i'd) 1 = d (ei - (ei ...

DIFFERENTIATING A CONVOLUTION WITH THE PARAMETRIX 247 (I-3) Estimating^82 ~Nj -^82 PNj. We shall show that fJx't,,uxx ax1axz ( ...

248 23. HEAT KERNEL FOR STATIC METRICS obtain (23.105) 1 fPPN .. (x, z, t - s) G (z, y, s) dμ (z) M 8x18x1 1 8PN 8G = - --j (x,z ...

3. DIFFERENTIATING A CONVOLUTION WITH THE PARAMETRIX 249 for s E [O, t - 8 (c)]. We conclude from splitting the RHS of (23.108) ...

250 23. HEAT KERNEL FOR STATIC METRICS where t* E (t, t + h) also depends on x, y, s. We claim that by taking the limit as h '\i ...

ASYMPTOTICS OF THE HEAT KERNEL FOR A STATIC METRIC 251 where a E G, 1) and where we used (23.102) and (23.106). From the above ...

(^252) 23. HEAT KERNEL FOR STATIC METRICS of the metric has the expansion (see formula (23.139) below) <let (gkc (x)) = 1-5kC ...

4. ASYMPTOTICS OF THE HEAT KERNEL FOR A STATIC METRIC 253 since J'VrJ^2 = 1. On the other hand, the space derivative of (23.115) ...

254 23. HEAT KERNEL FOR STATIC METRICS we have (23.121) .. 8 (al/2) -lJ. 8xJ = (bij + ~Ripqj (y) xPxq + 0 (r^3 )) x (~Rjq (y) xq ...

4. ASYMPTOTICS OF THE HEAT KERNEL FOR A STATIC METRIC 255 Now we compute the expansion for </>1 up to first order. From (2 ...

256 and HEAT KERNEL FOR STATIC METRICS Llx ( d^2 ( x, y)) L,1;!=0 Llx<faktk I L,1;!=0 \7 xtPk tk 1 2 Llx log H N = - + N - ...

ASYMPTOTICS OF THE HEAT KERNEL FOR A STATIC METRIC 257 More generally, for x near y and t near 0, we have ( f) ")(l H n 1 ( 4 ...

258 23. HEAT KERNEL FOR STATIC METRICS Then Let H be the heat kernel on a closed Riemannian manifold (Mn, g). From (23.132)-(23. ...

SUPPLEMENTARY MATERIAL: ELEMENTARY TOOLS 259 in (23.36). Using (23.127) and (23.128), we then have near t = O that^13 JM rJ (l ...

260 then HEAT KERNEL FOR STATIC METRICS det M ( 8) = 1 + 8 · tr (A) + 8^2 (tr ( B) + t tr^2 (A) - t tr (A^2 )) + 83 ( tr (A) ( ...