1547671870-The_Ricci_Flow__Chow

206 6. THREE-MANIFOLDS OF POSITIVE RICCI CURVATURE

where the constants Ci, c~ depend only on m and n. For instance, the case
m = 2 corresponds to the explicit formula

OiojRke = \l/V'jRke

• 
  

  ( rfj \l pRkz + rfk VjRpe + rfe \ljRkp + r;k \liRpe + r;e \liRkp)

  
  


• 
  

  ( r;Pr; 1 Rkq + r{Pr;kRqe + rferJkRqp + rfkfJeRpq)

  
  


• 
  

  ( air;kRpe + oir;eRkp).

  
  

Applying Corollary 6.47 and estimate (6.50), we see that

By the inductive hypothesis, we may assume that l8t8Pgl, hence laP Rel, has

been estimated for all 0 ::::; p < m - 1. By (6.51), this implies in particular

that loirl has been bounded for all 1 ::::; i ::::; m - 2. So to finish the proof, it
will suffice to estimate I am- l r I· We shall accomplish this by taking a time
derivative and integrating. By equation (6.1), we have

So by the inductive hypothesis and Corollary 6.50,

m- 1

::::; CL 1 am-1-igl iai\l Rei

i = O

m- 1

(6.52) ::::; c L loiVRc!.

i = O

Here we used the fact that bounds on g and its derivatives induce bounds

on the derivatives of g-^1 via the identity