1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

THE £-FUNCTION ON EINSTEIN SOLUTIONS AND RICCI SOLITONS 337 Since by (7.103), 2fR(O) ~ R(O) -l n R (r) ' we may rewrite (7.105 ...

338 7. THE REDUCED DISTANCE For example, we may take ( M^2 , g ( r)) to be an evolving 2-sphere and (N^1 ' h ( T)) to be the lin ...

THE £-FUNCTION ON EINSTEIN SOLUTIONS AND RICCI SOLITONS 339 Given a path 'Y (r), its £-length is £, (1') = [ v'i' (Rb (7) '7) ...

340 7. THE REDUCED DISTANCE COROLLARY 7. 72. The variation of the £-length on a steady gradient Ricci soliton Re (go)+ \J\J Jo = ...

THE £-FUNCTION ON EINSTEIN SOLUTIONS AND RICCI SOLITONS 341 Setting O' ~ 2y'T, we have Those £-geodesics (i.e., the critical p ...

342 7. THE REDUCED DISTANCE 7.3. £function on a gradient shrinker. Let (Nn,h(t)), -oo < t < 1, be a shrinking gradient Ric ...

THE £-FUNCTION ON EINSTEIN SOLUTIONS AND RICCI SOLITONS 343 Hence ;:, ('Y) = r Vi(.!. Rh(l) (/3 ( r)) + r [/3 (r) + .!. (grad ...

344 7. THE REDUCED DISTANCE Following up on details related to the previous exercise, we have EXERCISE 7.80. In regards to (7.11 ...

8. .C-JAOOBI FIELDS AND THE £-EXPONENTIAL MAP 345 On the other hand, taking I to be a minimal £-geodesic with I (f) = q,^13 we h ...

346 7. THE REDUCED DISTANCE where L ('-y) denotes the length of I· Integrating by parts on (7.119), we may express this as I (V, ...

8. .C-JACOBI FIELDS AND THE .C-EXPONENTIAL MAP 347 Thus we have a linear second-order ODE for the .C-J acobi field Y ( r) , call ...

348 7. THE REDUCED DISTANCE We now rewrite the £-Jacobi equation in a more natural way in view of the space-time geometry associ ...

8. .C-JACOBI FIELDS AND THE £-EXPONENTIAL MAP 349 we have D_g,_ (Rc(Y) + \i'yX - I_y) dr 2T = (! Re) (Y) + 2 (V' x Re) (Y) - 2 R ...

350 7. THE REDUCED DISTANCE 8.2. Bounds for £-Jacobi fields. Let c: > 0 and let 'Ys : [O, f] ----t M, s E ( -c:, c:) , be a s ...

8. .C-JACOBI FIELDS AND THE .C-EXPONENTIAL MAP 351 By (7.69), since (7.126) holds and Y (0) = 0, we have (oic) ('Y) - 2v1f ( x, ...

352 7. THE REDUCED DISTANCE REMARK 7.89. Note that for Euclidean space, £-Jacobi fields satisfy IY(r)l^2 = const·r. In particula ...

£-JACOBI FIELDS AND THE £-EXPONENTIAL MAP 353 EXAMPLE 7.90 (£-exponential map on a Ricci fl.at solution). To get a feel for th ...

354 7. THE REDUCED DISTANCE we have (^1) im-d. d{J ( ) p=1mpr l' 2 -d"( ~ d v ( pr=^2 -) v p---tO p p---tO T independent of 7'. ...

8. .C-JACOBI FIELDS AND THE £-EXPONENTIAL MAP 355 COROLLARY 7.93. Fix f E (0, T) and consider the £-exponential map Dr exp : TpM ...

356 7. THE REDUCED DISTANCE That is, if Ty < T, then 'TV is the first time the £-geodesic ')'V : [O, T) -+ M stops minimizing ...