1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

92 2. KAHLER-RICCI FLOW

LEMMA 2.63 (Linear algebra). There exist unitary vectors rl, ... , rN E
en with the property that, as Hermitian symmetric positive definite matrices,

N

(2.87) (i^3 (y)) = L av(Y )rv Q9 rv.

v=l

Here av(Y) E IR and JA :S av(Y) :SAA for some constant A > 0. Moreover

we may assume that the first n vectors ri, ... , r n form a unitary basis of en.

REMARK 2.64. Let { ei} ~=l denote the standard basis for <Cn and write
rv ~ 2:~ 1 (rv)i ei for each v. By (2.87) we mean that

N

(i1(y)) = Lav(Y) (rv)i (rv)j.

v=l

EXERCISE 2.65. Prove the above linear algebra lemma.

Define

Wv ~ Hess(u)(rv, rv) = Ui](rv)i(rvk

Now we let Mv(s) and mv(s) denote the quantities M(s) and m(s) defined
by (2.83) using Wv instead of w. Then (2.86) implies

N

(2.88) L av(Y)(wv(Y) - Wv(x)) :S h(y) - h(x).

v=l

Choosing x E B(2R) to be a point where w1(x) = m 1 (2), this in particular
implies

al(Y)(w1(Y) - w1(x)) :S h(y) - h(x) + L av(Y)(wv(x) - wv(Y))

v;:::2

and hence

w1(y) - m1(2) :S C(A, A) (RJJV'hJJco + L(Mv(2) - wv(Y))) ,

v;:::2

where we used al(Y) :2: JA, and av(Y) :SAA and wv(x) :S Mv(2) for v;: 2.
Thus
1

( R~n f,,(R) ( 1111 (y) - m1 (2) )P dy) ' <: C(A, A)RllV hlloo

1

(2.89) + C(A, A) L (R;n r (Mv(2) - Wv(Y))Pdy) p

v;:::2 j B(R)