1547671870-The_Ricci_Flow__Chow

204 6. THREE-MANIFOLDS OF POSITIVE RICCI CURVATURE

Since Mn is compact, Jt I^8 Z;J (x, T) I dT .. 0 as t / T uniformly in
g(t)
x E Mn. Hence the function f : T Mn .. JR defined by

f (x, V) ~ lim 9(x t) (V, V)

t-+T '

exists and is continuous in x E Mn and V E TxMn. By polarization, we
can define a (2, 0) -tensor 9(x,T) on Mn by

1

9(x,T) (V, W) = 4 [f (x, V + W) - f (x , V - W)] ,

noting that 9(x,T) (V, W) = limt->T 9(x,t) (V, W). The uniform bounds on
9 ( t) ensure that

e-c 9(x,o) (V, V) ::; f (x, V) ::; ec 9(x,O) (V, V),

hence that 9(x,T) is a Riemannian metric. D

In particular, the lemma implies that if (Mn, 9 ( t)) is a solution of the

Ricci fl.ow satisfying a uniform curvature bound on a finite time interval

[O, T), then all metrics in the family {9 (t) : 0 ::; t < T} are uniformly equiv-

alent.

COROLLARY 6.50. Let (Mn,9(t)) be a solution of the Ricci flow. If

there exists a constant K such that IRcl ::; K on the time interval [O, T],

then

e-^2 KT 9 (x, 0) ::; 9 (x, t) ::; e^2 KT 9 (x, 0)

for all x E Mn and t E [O, T].

We are now ready to prove the proposition.

PROOF OF PROPOSITION 6.48. Since Mn is compact, it is covered by

a finite atlas in which we have uniform estimates on the derivatives of the

local charts. Henceforth, we fix such a chart <p : U ____.. JR.n. Since g and the

connection '\7 are fixed, it will suffice to prove that for each m E N, the

ordinary derivatives of 9 of order m satisfy an estimate

for all x E U and t E t E [O, T),

where the norm I· I = I· 1, 5 is taken with respect to the Euclidean metric 15 in U ,

and Cm depends on only on m , n, K, T, and 90· Adopting this point of view,

we shall in particular regard r as a tensor in U , namely as the difference of

the Levi-Civita connection of 9 and the background (fl.at) connection in U.

The proof will be by complete induction on m EN.

In the estimates which follow, C will denote a generic constant that may

change from one inequality to the next, but which depends only on m, n,

K, T, and 90 · Choose {3 = {3(K,T) so that 0 < {3 < min{KT,1}. By

Corollary 6.50, we have uniform pointwise bounds from above and below for