1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

FIRST VARIATION OF £-LENGTH AND EXISTENCE OF £-GEODESICS 297 LEMMA 7.15 (.C-First Variation Formula). Given a 0^2 -family of c ...

298 7. THE REDUCED DISTANCE 3.2. The £-geodesic equation. The £-first variation formula leads us to the following. DEFINITION 7. ...

FIRST VARIATION OF £-LENGTH AND EXISTENCE OF £-GEODESICS 299 EXAMPLE 7.20 (£-geodesics on Einstein solutions). Let (N 0 , ho ( ...

300 7. THE REDUCED DISTANCE SOLUTION TO EXERCISE 7.21. Consider a maximal Einstein solution to the backward Ricci flow h (r), TE ...

FIRST VARIATION OF £-LENGTH AND EXISTENCE OF £-GEODESICS 301 EXERCISE 7.22. Estimate the speed of an £-geodesic for a solution ...

302 7. THE REDUCED DISTANCE equation: d2 ';JO ~ (-o ) d';ji d';jj d2 ((12) 0 (- ) (Cl) 2 dCl2 + ~ rij^0 (3 dCl dCl = dCl2 4 + I ...

FIRST VARIATION OF C-LENGTH AND EXISTENCE OF C-GEODESICS 303 {3° - (0) = 0 and d~o dri" (er) =f. 0 for er > 0, then {3° - ( ...

304 7. THE REDUCED DISTANCE Since 72 < T, by (7.27) and the Bernstein-Banda-Shi derivative estimate (Theorem Vl-p. 224), ther ...

FIRST VARIATION OF £-LENGTH AND EXISTENCE OF £-GEODESICS 305 curvature. Given a space-time point (p, T1) E M x [O, T) and a ta ...

306 7. THE REDUCED DISTANCE Letting O'j ~ 2..Jij and /Ji (O') ~'Yi (0'^2 /4), the above formula says 1 0'2 Id/Ji 12 dO' < c. ...

GRADIENT AND TIME-DERIVATIVE OF THE £-DISTANCE FUNCTION 307 (7.50) Recall that (3 (O") is defined by (7.18). By Lemma 7.13(ii) ...

308 7. THE REDUCED DISTANCE where 0 = 2 e4Coc:+6C 0 T (e2C 0 T 2 d~(T2) (p, q) + 2n0o T + c'#,T) (^3). 4T2 3 2 1205 (using la - ...

GRADIENT AND TIME-DERIVATIVE OF THE L-DISTANCE FUNCTION 309 is a piecewise smooth path from p to q2. We estimate, using I~~ (T ...

310 7. THE REDUCED DISTANCE 4.2. Gradient of L. We compute the gradient of L via the first vari- ation formula for £. Since L ( ...

GRADIENT AND TIME-DERIVATIVE OF THE £-DISTANCE FUNCTION 311 LEMMA 7.32 (Gradient of L formula). The spatial gradient of the L- ...

312 7. THE REDUCED DISTANCE ~~ (q, 7) ~~ (q, 7). If L (q, ·) is not differentiable at 7, then by writing ~~ (q, 7) = y'T ( R ('! ...

THE SECOND VARIATION FORMULA FOR .C AND THE HESSIAN OF L 313 L defined in subsection 4.2 of this chapt~r. At a point q where L ...

314 7. THE REDUCED DISTANCE On the other hand, we compute d dT (\lyY,X) = (\lx\lyY,X) + (\lyY, \lxX) ~~ (\1 y Y, X) + ( ( :T \l ...

THE SECOND VARIATION FORMULA FOR L AND THE HESSIAN OF L 315 where we used the £-geodesic equation (7.32) to get the last equal ...

316 7. THE REDUCED DISTANCE Hence (7.62) and Y (0) = 0 imply 1 - 2 (6f£) (1) -yl;; \7yY,X) + VTRc (Y, Y)I; = foT VT ( (% 7 Re+ 2 ...