1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

WEAK SOLUTION FORMULATION 377 Note that lims---tO+ ftn:n d~ (\l<p · v) dμ = 0 (since n pairs of opposing bound- ary terms c ...

378 7. THE REDUCED DISTANCE (i) The inequality of 2 n or -.6.P + IV Pl - R + 2r 2:: 0 holds in the weak sense on M x (0, T), i.e ...

6. Notes and commentary LEMMA 7.130 (Integration by parts inequality for£). We have (7.148) JM (il£ - l\7£1^2 ) e-£dμ ~ 0. PROOF ...

380 7. THE REDUCED DISTANCE (Mn, g ( T)): a solution to the backward Ricci flow, usually defined for T E [O, T] , complete, and ...

Chapter 8. Applications of the Reduced Distance. Now ... the basic principle of modern mathematics is to achieve a complete fusi ...

382 8. APPLICATIONS OF THE REDUCED DISTANCE EXERCISE 8.1. Show that if ( JVin, g) is a complete noncom pact Rie- mannian manifol ...

REDUCED VOLUME OF A STATIC METRIC 383 where we used JV'dJ = 1 a.e. and the Laplacian comparison theorem, i.e., d.b..d::::; n - ...

384 8. APPLICATIONS OF THE REDUCED DISTANCE That is, 82 ()()2 ( ()2 2 ) = 1 - 27r.^0 7r• The square torus is also a nice concret ...

REDUCED VOLUME OF A STATIC METRIC 385 Let A (s) denote the volume of the geodesic (n - 1)-sphere of radius s cen- tered at p. ...

386 8. APPLICATIONS OF THE REDUCED DISTANCE then 1-.'!.!! 0 < V - ( g, "'ar 2) S K,^2 K, (47rt^12 + -. wn Hence if a< 2/n, ...

REDUCED VOLUME FOR RICCI FLOW 387 2.1. Volumes of geodesic spheres in M. We motivate the definition of the reduced volume by c ...

388 8. APPLICATIONS OF THE REDUCED DISTANCE is roughly (since the sphere has small curvature for N large) the volume of the hype ...

REDUCED VOLUME FOR RICCI FLOW 389 See Lemma 8.16(ii) below for why V (7) is well defined even when Mis noncom pact. In the cas ...

390 8. APPLICATIONS OF THE REDUCED DISTANCE formulas to be considered rigorously later. Let X =\!£.Again, provided we can differ ...

REDUCED VOLUME FOR RICCI FLOW 391 We will use the convention £Jv (r) ~ 0 for r 2: rv. We can then write the reduced volume as ...

392 8. APPLICATIONS OF THE REDUCED DISTANCE (ii) For any VE TpM and 0 < 7 < T, (8.23) ( 47r 7 )-n/2 e-£bv(r),r) £ Jv ( 7 ) ...

REDUCED VOLUME FOR RICCI FLOW 393 Since n(p,0) (r1) = n (r1) ::::) n (r2) for r1 < r2, we have limT->0+ Xn(T) = 1, where ...

394 8. APPLICATIONS OF THE REDUCED DISTANCE for all TE [Ti, T 2 ]). This implies f, is C^00 on M x [Ti, T2] and (8.26) holds on ...

REDUCED VOLUME FOR RICCI FLOW 395 Then for any 71 < T2, PROOF. By the definition of the .C-Jacobian we know that for any L^ ...

396 8. APPLICATIONS OF THE REDUCED DISTANCE PROOF. We justify the differentiation under the integral sign in equality (8.17). Co ...