1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

is4 4. PROOF OF THE COMPACTNESS THEOREM

By the previous lemma, cm(μi, ... ,μk) {qi, ... , qk} is the unique solution of the

equation

G (q) ~ Gqi, ... ,qk,μ1, ... ,μk (q) = 0.

Consider the partial derivative

k

· \7 qG = L μi \7 q exp;i qi : TqM ---+ TqM·

i=i

By Lemma 4.45, \7 qG is positive definite with smallest eigenvalue being

bounded from below by a constant depending only on Co and μi, ... , μk. It

follows from the implicit function theorem that the unique solution

Cm(μ 1 , ... ,μk) {qi,···, qk}

is continuous in qi, ... , qk and μi, ... , μk.

To see that the \7~\7~-covariant derivatives of cm(μi, ... ,μk) {qi, ... , qk}

are bounded, we compute the other partial derivatives of G:

Hence

(4.25)

\7 qiG = μi\7 qi expq-i q,i, ~G^8 = expq -i qi.

uμi

( \7 qiG, f}~j G) + \7 qG · (\7 qi cm {qi, ... , qk}, \7 μj cm {qi, ... , qk}) = 0,

where q =cm {qi, ... , qk}. Thus

(\7 qi cm {qi, ... , qk}, \7 μj cm {qi, ... , qk}) = - (\7 qG)-i ( \7 qiG, f}~j G).

This and Proposition 4.48(i) implies (4.24) when lal + l,BI = 1.

To bound the higher derivatives of cm {qi, ... , qk} , we argue inductively

on the order of the derivative lal· + l,BI. We take the appropriate derivatives

of (4.25) of order lal + l,BI - 1 with respect to qi, ... , qk and μi, ... , μk so

that Y'qG · \7~\7~ cm(μi, ... ,μk) {qi, ... , qk} appears in the resulting equality.

Then

\7~\7~ Cm(μ 1 , ... ,μk) {qi,···, qk}

can be expressed in terms of \7~~ expq-i qi with Zi :S: lal + l,BI, \7~;\7 q expq-i qi

with £2 :S: lal + l,BI, \7~^1 \7~^1 cm(μ 1 , ... ,μk){qi, ... ,qk} with lail + l,Bil :S:

lal + l,BI - 1, and ('VqG)-i. Now it is easy to see from Proposition 4.48(i)

that [\7~\7~ cm(μ 1 , ... ,μk) {qi, ... , qk}[ are bounded by constants Clal+l,61+1 de-

pending on n, inj (p), lal + l,BI, and Co, ... , Clal+l,61+1·

(ii) When qi, ... , qk ---+ q*, by Lemma 4.50, we have 8 ~i G ---+ 0. By

Proposition 4.48 (ii) we have \7 qi G ---+ μi id and \7 qG ---+ - ( I::~=i μi) id.