1547671870-The_Ricci_Flow__Chow

54 2. SPECIAL AND LIMIT SOLUTIONS

where

(2.58)

K

P = ( n - 1) log L - 2 L (log L - 1)

and

(2.59)

K2

Q = n - 1 - L 2 (log L - 1) - F.

PROOF. Straightforward computation. D

Rather than work with F directly, we shall consider a related quantity

F, which is invariant under simultaneous rescaling of the metric g f---t >.g and

time t f---t >.t.

LEMMA 2.39. Let g (t) : 0 ::; t < T be a maximal solution of the Ricci

flow having the form (2.43) such that IVisl ::; 1 and R 2: 0. For all t E [O, T)

and x E W (t), the scaling-invariant quantity

A K

F ~ L [log L + 2 - log (Lmin (O))]

satisfies

sup F (., t) ::; max {n -1, sup F (., o)}.

W(t) W(O)

(2.60)

PROOF. We first consider the case that Lmin (0) 2: e^2. Then by Corollary

2.26, we have Lmin (t) 2: e^2 for as long as the solution exists. We will apply

the maximum principle to equation (2.57) in the region where F 2: n - 1,

noting that K > 0 in this region as well. Our assumption that 0 ::; R =

n [-2K + (n - 1) L] implies that

K < n - l L.

• 2

Using L > e^2 > e, we conclude that the coefficient P in (2.58) satisfies

P 2: ( n - 1) log L - ( n - 1) (log L - 1) = n - 1.

Thus when K > 0 and L > e^2 , equation (2.57) implies the differential

inequality

(

logL- 1)

Ft :S b..F + 2 LlogL LsFs + 2QK.

But if F 2: n - 1, then the coefficient Q in (2.59) satisfies

K2

Q::; - L 2 (logL - 1)::; 0.

Hence

(

logL-1)

Ft :S b..F + 2 L log L LsFs.

We conclude that if Lmin ( 0) 2: e^2 , then sup F ( ·, t) cannot increase whenever

it exceeds n - 1.