1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

4. THE LIMIT MANIFOLD (M~,g 00 ) 169

Notice that with respect to the coordinates ( Ef3, Hf) the map q/{; ( x) ,

where a i= 0, can be expressed as

~ex ..:... ex /3 _ X^0 Jk /30 · n/,ex 'f/^0 J(3ex k

n,(3 -,--¢k^0 Hk - 0 130 130 /3"!'

1/J^0 Jk + I":o<"!::;A(r) X^0 Jk · 1/J"I^0 Jk

so with respect to the coordinates ( Ef3, Hf) , we have that ¢k,,/3 converges

to a function ¢~,/3 defined by

When a = 0, we have

~o _,_ ~o f3 _ n/,0 'f'^0 J/30 k

'f'k,/3 -,--'f'k^0 Hk - 0 130 130 /3"!'

1/J^0 Jk + I":o<"!::;A(r) X^0 Jk · 1/J"I^0 Jk

0 0. n/,0 'f/^0 .Joo Tf30

¢k,(3 ---+ ¢00,/3 ::::;= 0 (30 (30 f3"! ·

1/J^0 Joo + I":o<"f::;A(r) X^0 Joo · 1/J"I^0 Joo

The definition of Fke;r is as follows. For x E B (Ok, r) , we define

(4.11) Fke;r (x) ~cm { F~e (x), F"fe (x), ... , F:C(r) (x)} E Me

to be the center of mass using the weights ¢k, ( x); by the choice of balls
in Lemma 4.18 we can apply Proposition 4.53 and conclude the existence
of Fke;r when k, £ are large enough. The map Fke;r is smooth with all its
derivatives !'VP Fke;rl bounded by constants Cp+l independent of k. From the
construction of the weights ¢k, we have Fke;r (Ok)= Oe. Note by Proposition
4.53 and the definition of the center of mass, Fke-r satisfies
(i) Fke;r (x) is the minimizer y E Me of '
A(r)
fx (y) = L ¢k, (x) d~£ (y, Fk£ (x)),
ex=O
(ii) Fke;r (x) is the solution y E Me of
A(r)

(4.12) L ¢k, (x) exp;^1 Fk£ (x) = 0,

where the exponential map is with respect to ge. With respect to the coor-

dinates ( Ef3, Hf) , equation ( 4.12) can be written in Ef3 as

L ¢k, o Hf (X) exp-Fk£;roHk^1 f3( X ) Fk£ o Hf (X) = 0.

ex::;A(r)