1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

THE SECOND VARIATION FORMULA FOR LAND THE HESSIAN OF L 317 we have 1 7 2y'T IVxY +Re (Y)l^2 dr = 1 7 2y'T [vxY +Re (Y) - 2 ~Y[ ...

318 7. THE REDUCED DISTANCE LEMMA 7.39 (£-Second variation under (7.66)). If f E (0, T), Yr E T'Y(r)M and Y is a solution to (7. ...

THE SECOND VARIATION FORMULA FOR £ AND THE HESSIAN OF L 319 If L (·, f) is not C^2 at q, the above inequality is understood in ...

320 7. THE REDUCED DISTANCE Plugging the above estimate into (7.70) and using IY (7)1^2 = ¥ IY (f)l^2 , we get D 5.3. Laplacian ...

THE SECOND VARIATION FORMULA FOR .C AND THE HESSIAN OF L 321 Using the Einstein summation convention (expressions with i repea ...

322 7. THE REDUCED DISTANCE As before, if L (·, T) is not C^2 at q, by our convention the above inequality is understood in the ...

EQUATIONS AND INEQUALITIES SATISFIED BY L AND £ 323 LEMMA 7.43 (L-distance and trace Harnack quadratic). Let f E (0, T). Suppo ...

324 7. THE REDUCED DISTANCE that is, (7.85) (:r +il) (L-2nr) :S 0. Now we prove a lemma which describes the limiting behavior of ...

EQUATIONS AND INEQUALITIES SATISFIED BYLAND£ 325 PROOF. (i) From (7.85) we have :r (L-2nr) +L1(L-2nr) ~O. If L ( ·, ·) were C^ ...

326 7. THE REDUCED DISTANCE Hence b..L(q 7 , r) ?:: 0. We have d+h(-)<l. L(q7,r+s)-L(q7,r) -d T _ imsup T s-+O+ S 1 :::; im. ...

EQUATIONS AND INEQUALITIES SATISFIED BY L AND e 327 LEMMA 7.50 (Reduced distance - partial differential inequalities). At (q, ...

328 7. THE REDUCED DISTANCE may write formula (7.65) at smooth points off as (7.96) ( V'V'f +Re- 2 ~g) (Y, Y) (f) =- 2 ~fo 7 y'T ...

EQUATIONS AND INEQUALITIES SATISFIED BYLAND .e 329 SOLUTION TO EXERCISE 7.55. We compute, using (7.94), -^8 · ( 8£ £ d1 ) 8 (2 ...

330 7. THE REDUCED DISTANCE 6.4. The growth of the reduced distance function f. The growth off, in particular the lower bound of ...

EQUATIONS AND INEQUALITIES SATISFIED BY L AND £ 331 (ii) From (7.94) we have at (q, T), 1 0£1 <@J Jfl ~IV£l2. OT - 2 + 2T + ...

332 7. THE REDUCED DISTANCE since IY (7)1^2 = :¥ IY (r)l^2. Hence by (7.70), (Hess(q,·r) L) (Y (r), Y (r)) :S~IY(r)l2 foT vr(1+T ...

EQUATIONS AND INEQUALITIES SATISFIED BYLAND .e 333 where C = 1 + ~· If T = oo, then for any (q, f) EM x (0, oo) . 3. 1vc1^2 (q ...

334 7. THE REDUCED DISTANCE an exhaustion {Uk}kEN of M 00 by open sets with Poo E Uk and diffeomor- phisms <I>k : Uk ~ Vk ...

THE £-FUNCTION ON EINSTEIN SOLUTIONS AND RICCI SOLITONS 335 The second equality above needs uniform bound ..Ji I~:; 1;k (T) :: ...

336 is THE REDUCED DISTANCE d;(O) (p, q) 1; </> (T)-l dT and the minimizer is given by a minimal geodesic 'Y with speed ...