1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

CONSTRUCTION OF THE PARAMETRIX FOR THE HEAT KERNEL 221 (1) The defining equation (23.11) implies the following first-order lin ...

222 23. HEAT KERNEL FOR STATIC METRICS By induction on k ::::;: 0, we see that the functions k defined recursively by (23.26) ar ...

CONSTRUCTION OF THE PARAMETRIX FOR THE HEAT KERNEL 223 in the second line) (23.29) ofo: (DxHN) (x, t; y, u) e k =LL coeff ·o{o ...

224 23. HEAT KERNEL FOR STATIC METRICS on Minj(g) x JR}. Moreover, if N > ~ + k + 2£, then (23.33) af\J~ (DxHN) (x, t; y, u) ...

CONSTRUCTION OF THE PARAMETRIX FOR THE HEAT KERNEL 225 a cutoff function to obtain a parametrix for the heat operator. Define ...

226 23. HEAT KERNEL FOR STATIC METRICS (iii) In the intermediate region (Minj(g) - Minj(g)/s) x IRt by (23.38) and (23.11), we h ...

CONSTRUCTION OF THE PARAMETRIX FOR THE HEAT KERNEL 227 is bounded independent of (p, y) in any given compact subset](, of Xx M ...

228 23. HEAT KERNEL FOR STATIC METRICS (Minj(g) - Minj(g)/s) X lRf, . -(n/2) ( d^2 (x,y)) ( (inj(g)/8) 2 ) JHNI (x, t, y, u) ~ C ...

EXISTENCE OF THE HEAT KERNEL ON A CLOSED MANIFOLD 229 their (space-time) convolution is given by (23.46) (F * G) (x, y, t) ~la ...

230 23. HEAT KERNEL FOR STATIC METRICS For basic properties and results about general convolution transforms, see Hirschman and ...

2. EXISTENCE OF THE HEAT KERNEL ON A CLOSED MANIFOLD 231 THEOREM 23.16. For N > ~ and GN as in (23.55), the function (23.56) ...

232 23. HEAT KERNEL FOR STATIC METRICS STEP 3. H is the fundamental solution. Recall from Proposition 23.12 that we have for any ...

2. EXISTENCE OF THE HEAT KERNEL ON A CLOSED MANIFOLD 233 as t--+ 0, uniformly for allx, y EM. EXERCISE 23.18. Show that for k, i ...

234 23. HEAT KERNEL FOR STATIC METRICS PROOF. Since qN E c^0 (Nl x M x (O,oo)) and qN(x,y,t) is C^00 in x and t, the issue is at ...

EXISTENCE OF THE HEAT KERNEL ON A CLOSED MANIFOLD 235 converges for all t E lR+, we have the following. COROLLARY 23.21 (Conve ...

236 23. HEAT KERNEL FOR STATIC METRICS where we also used (t - s)N-~ ~ tN-~ and^8 i t k(N-1'+1)-l tk(N-~+l) s 2 ds =. o k (N - ~ ...

EXISTENCE OF THE HEAT KERNEL ON A CLOSED MANIFOLD 237 From this we see that for F satisfying the above hypothesis and k 2:: 2, ...

238 23. HEAT KERNEL FOR STATIC METRICS Hence 00 L of a;: ( (DxPN )*k) (x, y, t) k=l converges absolutely and uniformly on K x K ...

DIFFERENTIATING A CONVOLUTION WITH THE PARAMETRIX 239 We leave the proof as an exercise or see Theorem 1 on pp. 4-6 of [61] wh ...

240 23. HEAT KERNEL FOR STATIC METRICS B (x, inj (g) /2). Hence, regarding the space integral in (23.82), by (23.83) and a > ...