1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

148 30. TYPE I SINGULARITIES AND ANCIENT SOLUTIONS

STEP 3.

(30.57a)

(30.57b)

Define

The ODI f or u (s) in (30.48). By taking z = f3(s) in (30.56) we have

1

u'(s)::; ylW=tJA+Bu(s),

w-t

u (o) = e (q, t).

1 ( B s )

2

U(s)=:= B 2vw=t+JA+Be(q,t)

A

B '

which is the solution to the corresponding ODE

1

(30.58a ) U'(s)= ~JA+BU(s),

yw-t

(30.58b) u (o) = e (q, t).

By the ODE comparison theorem , we have u(s)::; U(s) for all s E [O,dg(t) (q,x)].

In particular, by takings= dg(t) (q, x), we obtain

f (xt)<_!_(Bdg(t)(q, x)+ IA+Be( t))2 A

' - B 2Jw - t v q, B

Bd~(t) (q, x ) A

::; 2 ( w - t) + B +^2 e ( q' t) '

which is (30.46).

STEP 4. Proof of (30.47). Taking q = p and using R (p, t) ::; :,

2

!~, we obtain

(using the constant path as the test path for e)

(30.59) f(p, t)::; ~ lw ~R (p , i ) di

2 w - t t

::; n^2 M.

Applying this to (30.46) yields (30.47). 0

We also have bounds for the first derivatives of e in both space and time.

COROLLARY 30.16 (Gradient and time derivative bounds fore of T ype I solu-

tions). Under the hypoth esis of Lemma 30.15, there exist constan ts C , D , E , F < oo

depending on n , c E (0, w2°'), and M such that

(30.60) IV£1 (x, t) ::; _C_ + D dg(t) (p , x)

ylW=t w-t

and

(30.61)

on M x [a + c, w).

l

ael() E Fd~(t)(p,x)

• xt <--+---~
  at ' -w -t (w - t)^2

PROOF. (1) Applying (30. 4 7) to (30.56), we obtain

2A + 2Bn2 M B^2 d~(t) (p , x)

-----+ 2

w - t 2 (w -t)

(30.62) 1ve1 (x, t)::;

This proves (30.60) with C = J2A + 2Bn^2 M and D = ~' where A and Bare
given by (30.54)- (30.55).