1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

5. THE SECOND VARIATION FORMULA FOR .C AND THE HESSIAN OF L 321

Using the Einstein summation convention (expressions with i repeated

are summed from 1 ton), we now simplify L,~ 1 H(X,Ei) (r):

i=l

= -2 ( :r Re) (Ei, Ei) - \7 Ei \7 EiR + 2 IRc (Ei) 12 - ~Re (Ei, Ei)

• 2 (R (Ei, X) Ei, X) + 4 (\7 Ei Re) (Ei, X) - 4 (\7 x Re) (Ei, Ei)
  8 'T 'T 2
  = -2-;;;-[Re (Ei, Ei)] + 4Rc (\7 xEi, Ei) - -=l::i.R + 2-= IRcl
  U'T 'T 'T
  lr r r r


• --=R + 2-= Re (X, X) + 4-= div (Re) (X) - 4-= \7 xR
  'T 'T 'T 'T 'T

= -2! (~R) + 4Rc (-Re (Ei) + 2 ~ Ei, Ei) - ~l::i.R + 2~ 1RcJ

2

1 'T 'T

• -=R + 2-= Re (X, X) - 2-= \7 xR
  'T 'T 'T

= ~ (-2

8

R - l::i.R- 2 JRcl^2 - R + 2Rc (X, X) - 2\lxR).

'T OT 'T '.

Recall from (Vl-p.274) that Hamilton's trace Harnack expression is

EJR R

(7.73) H (X) ~ H (X) (r) ~ -~ - 2\7R · X + 2Rc (X,X) - -

u'T 'T

(in (Vl-p.274) let t = -r and replace X by -X). Using ~~ = -l::i.R -

2 IRcl^2 , we get
n

(7.74) "'£H (X,Ei) (r) = ~H (X).

i=l 'T

Hence

1

7 7 3/2 n

l::i.L (q,T) ::::; - -_-H (X) dr + '= - 2v'f-R (q, f)

0 'T y'T

1 n '=

= --=K + '= - 2vfR(q,f),

'T v 'T

where

(7.75)

We call K the trace Harnack integral. Dropping the bars on the r's, we
have the Laplacian comparison theorem for L.

LEMMA 7.42 (Inequality for l::i.L). Given r E (0, T) and q E M, let

I: [O, r] --t M be a minimal £-geodesic joining p to q. Then

1 n

(7.76) l::i.L (q, r) ::::; --K + r:; - 2yfTR.

'T v 'T