1547671870-The_Ricci_Flow__Chow

202 6. THREE-MANIFOLDS OF POSITIVE RICCI CURVATURE

can accomplish this by the following strategy: when we have an upper bound
on IRml which is uniform in time, we can apply Theorem 6.46 at large times
by taking a fixed small step backwards in time. This technique lets us prove
the following consequence of the theorem.

COROLLARY 6.47. Let (Mn, g (t)) be a solution of the Ricci flow for

which the weak maximum principle holds. If there are /3 > 0 and K > 0

such that

IRm (x, t) l 9 ::; K for all x EM and t E [O, T] ,

where T > /3 / K, then there exists for each m E N a constant Cm depending

only on m , n, and min {.B, 1} such that

IVm Rm (x, t) l 9 ::; CmKl+'g' for all x E Mn and t E [ min .i:' l}, T].

PROOF. Let /31 ~min {/3, 1}. Fix to E [,Bi/ K , T ] and set To~ to-/3i/ K.

Let f ~ t - To, and let g (f) solve the Cauchy problem

8 -

otg = - 2Rc

g ( 0) = g (To).

Then by uniqueness of solutions to the Ricci fl.ow, g (f) = g (t +To) = g (t)

fort E [O, ,Bi/ K ]. Thus by hypothesis on the solution g (t), we have

I Rm (x, f) 19 ::; K

for all x E Mn and t E [O, ,Bi/ K]. Applying Theorem 6.46 with a = ,8 1 , we

get a family of constants Cm depending only on m and n such that

1

-m- I CmK

\7 Rm (x, f) 9 ::; fm/2

for all x E Mn and t E (0, ,Bi/ K]. Now when t E [ f_k, ~ J, we have

pn/2 ~ ,B"{1'/2Tm/2 K-m/2.

Taking t = /31 / K, we find in particular that

(

/2 - )

lvm Rm (x ' t O )I g -< 2m m/2 Cm Ki+m/2

/31

Since to E [,Bi/ K, T] was arbitrary, the result follows. D

The main objective of this subsection is to obtain the following gradient

estimates. They are stated with respect to a half-open time interval [O, T)

because one of their applications (moreover, the one of current interest) is

to help us study obstacles to long-time existence of the fl.ow.