1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

BOUNDS OF THE HEAT KERNEL FOR AN EVOLVING METRIC 361 Hence d~(x,y) c1e -cz(r-v). H (x, T; y, v) ~ 1/2 1/2. Vol 9 B9 (x, VT -v) ...

362 26. BOUNDS FOR THE HEAT KERNEL FOR EVOLVING METRICS for i = 1, 2. Using the semigroup property (26.41), which implies we com ...

HEAT BALLS AND THE SPACE-TIME MEAN VALUE PROPERTY 363 By the monotonicity formula (26.55) in the proof of Lemma 26.21, we have ...

364 26. BOUNDS FOR THE HEAT KERNEL FOR EVOLVING METRICS PROOF. Consider the function of r defined by the RHS of (26.83), which i ...

HEAT BALLS AND THE SPACE-TIME MEAN VALUE PROPERTY 365 EXERCISE 26.36 (Mean value inequality when Re 2: 0). Show that if (Mn, g ...

366 26. BOUNDS FOR THE HEAT KERNEL FOR EVOLVING METRICS Another useful way to describe the heat ball is Er(x, t) = {(y, s) E JR. ...

HEAT BALLS AND THE SPACE-TIME MEAN VALUE PROPERTY 367 Note that the condition t;:::: ~; is equivalent to Er(x, t) C ffi.n X [O ...

368 26. BOUNDS FOR THE HEAT KERNEL FOR EVOLVING METRICS (n - 1)-dimensional measure on 8Er,s(x, t). Since we have x-y "Vy'l/Jr= ...

HEAT BALLS AND THE SPACE-TIME MEAN VALUE PROPERTY 369 Its time slices are given by E;^1 (xo, to) =i= Er(xo, to) n { T (t) = T1 ...

370 26. BOUNDS FOR THE HEAT KERNEL FOR EVOLVING METRICS PROOF. (1) The integral is finite. Let (xo, to) EM x (0, T]. By (26.103) ...

HEAT BALLS AND THE SPACE-TIME MEAN VALUE PROPERTY 371 Noticing that the set Er,t is empty for t sufficiently close to -T, by t ...

372 26. BOUNDS FOR THE HEAT KERNEL FOR EVOLVING METRICS deduce :/ ~ ~ L (!,,,,, ( \O,~u + u ( 8 ti' + 1vM)) d1}t = -n 1°^1 ( 1/J ...

HEAT BALLS AND THE SPACE-TIME MEAN VALUE PROPERTY 373 Hence, from the absolute continuity of the function I(r), we have u(xo, ...

374 26. BOUNDS FOR THE HEAT KERNEL FOR EVOLVING METRICS where ii is the outward unit normal to 8D and where d& is the volume ...

HEAT BALLS AND THE SPACE-TIME MEAN VALUE PROPERTY 375 3.4. Mean value property under Ricci flow. The mean value property can a ...

376 26. BOUNDS FOR THE HEAT KERNEL FOR EVOLVING METRICS Hence, if u ~ 0, then I(r2) -I(r1) :S -1~ 2 r~l lr 'I/Jr (:t -.6) udμg(t ...

DISTANCE-LIKE FUNCTIONS ON NONCOMPACT MANIFOLDS 377 Given r > 0, t E (-oo, oo ), and O E (0, 1), we define the following pa ...

378 26. BOUNDS FOR THE HEAT KERNEL FOR EVOLVING METRICS PROPOSITION 26.49 (Distance-like functions with bounds on deriva- tives) ...

DISTANCE-LIKE FUNCTIONS ON NONCOMPACT MANIFOLDS 379 Step 1. We shall show that there exists a constant C 1 < oo depending o ...

3SO 26. BOUNDS FOR THE HEAT KERNEL FOR EVOLVING METRICS where VolK B(s) denotes the volume of the ball of radius s in the simply ...