1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

90 19. GEOMETRIC PROPERTIES OF K;-SOLUTIONS

this solution. One may simply apply Hamilton's matrix Harnack estimate

on each interval [c-,w), where c E (O,w), and then take c---+ o+.

Remark 19.30 may also lead one to ponder the following.

PROBLEM 19.31 (Continuity of the supM IRml (t) function). Suppose

that (Mn, g ( t)), t E (a, w), is a complete solution to the Ricci fl.ow such

that supM IRm (g (t))I < oo for each t E (a, w). Is supM IRm (§ (t))I a con-

tinuous function of t? A weaker question is to ask this under the additional

assumption that Rm (g (t)) 2:: 0.^10

2.3. A characterization of the ,,;-noncollapsed condition for an-

cient solutions.

In this subsection we shall discuss the following characterization of ,,;-

solutions in terms of the boundedness of the entropy of the fundamental

solution to the adjoint heat equation. At the end of §11.1 in [152], Perelman

wrote:

Let

(19.12)

We impose one more requirement on the solutions; namely,

we fix some ,,; > 0 and require that 9ij ( t) be ,,;-noncollapsed

on all scales (the definitions 4.2 and 8.1 are essentially equiva-

lent in this case). It is not hard to show that this requirement

is equivalent to a uniform bound on the entropy S, defined

as in 5.1 using an arbitrary fundamental solution to the con-

jugate heat equation.

denote Perelman's entropy functional and let

D* == _ !!_ - .6. + R

· at

denote the adjoint heat operator. Recall from (17.12) that under the

coupled system

a

-g = -2Rc

at '

D*u = 0,

dr = _ 1

dt '

where u ~ (41rr)-nl^2 e-f, we have

(19.13)! lVV(g (t), J (t), r (t)) = 2r JM IRc +\TV' f - 2 ~gl

2

udμ 2:: 0.

Now we may interpret §11.1 of [152] as saying the following.

(^10) Another weaker question is to ask only whether for every [,B, '1f!] c (a, w) we have
SUPMx[,6,,P] fRmf < oo.