1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

6. 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 + 2

Using (7.98), we get at (q, T) that

I

OT of I s Co+ 1£1^01 ( A )

2 T +--;- f +AT

If,+ ATI +AT 01 ( A )

S Co +

2

+ - f +AT

T T

s ( 61 + ~) (£ + (A+ c~ + ~:) T) ,

T Cr+ 2

where we have used f (q, T) +AT ;:::::: 0 for all (q, T) in the last inequality.

A 1 A Co+lA.
Thus (ii) follows from taking C1 = C1 + - 2 and A = A+ - 0 ,^21 • The lemma
1+2
is proved. D

From Lemma 7.41 we get

LEMMA 7.61 (Hessian of reduced distance). Suppose (Mn,g(T)), T E

[O, T), is a complete solution to the backward Ricci flow with bounded sec-

tional curvature. Fix To E (0, T). Given f E (0, To] and q EM, the Hessian

of the reduced distance function f (·, f) at q has the upper bound

c 2 1 + C2d;C'f) (p, q)

(7.99) Hess(q,T) f S

2

.;T + 27 '

where C2 is a constant depending only on n,To,T,supMx[O,T] IRml.

NOTATION 7.62. Let <p be a C^2 function in a Riemannian manifold

( M, g). By Hess <p ~ \JV <p S C we actually mean \JV <p S Cg. We shall

similarly abuse notation in other parts of this chapter.

6.5. Estimates for f when Rm;:::::: O. As we shall show below, when

(Mn, g ( T)) , T E [O, T), has bounded nonnegative curvature operator, there

are better estimates for Hess(q,7') £, IV£1^2 , and I g~ I· Let f E (0, T) and
q E M. Let 'Y : [O, f] -t M be a minimal £-geodesic from p to q and let
X (T) ~ ~:;'.. Let Y be a solution to (7.66). Hamilton's matrix Harnack
estimate implies

H (X, Y) (T) 2:: - (~ + T ~ T) Re (Y, Y) (T)

2:: -n.(~ + T ~ T) RIY (T)l

2

;:::::: -~ (1 + _T_) IY (r)l^2 R

T T-T