1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

2. THE 1£-NONCOLLAPSED CONDITION 91

CLAIM 19.32 (Entropy and A:-noncollapsing for ancient solutions). Let

(Mn,g(t)), t E (-oo,O], be a complete ancient solution to the Ricci flow

such that Rm(g(t)) 2: 0 and supM IRm(g(t))I < oo for each t E (-oo,O].

Then there exists A: > 0 such that g (t) is A:-noncollapsed at all scales if

and only if there exists C < oo such that for any fundamental solution

u: M x (-oo, to) --+IR+ of the adjoint heat equation

D*u = 0,

where limt--+(to)_ u (t) = bp 0 for some (po, to) EM x (-oo, O], we have

(19.14) sup IW (g (t), f (t), to - t)I :::::; C,

tE(-oo,to]

where u (t) = (47r (to - t))-n/^2 e-f(t) .11

PROBLEM 19.33. Prove this claim.^12

For comparison, consider the case of the heat equation on a static Rie-

mannian manifold (Mn, g).^13 Let

Wlin (f, t) = (47rt)-n/^2 JM (t IV fl^2 + f - n) e-f dμ

denote the linear entropy functional. If (Mn, g) is a closed Riemannian
manifold with nonnegative Ricci curvature and u (x, t) = (47rt)-n/^2 e-f(x,t)
is a solution to the heat equation, then (see Theorem 0.1 in [139])

:t w.. u ( t) , t) ~ -2t L (I"" / -;tgl' +Re(" 1, "n) u dμ ::; o.

In Proposition 3.2 of [139], one of the authors proved the following.

THEOREM 19.34. Let (Mn, g) be a complete noncompact Riemannian

manifold with nonnegative Ricci curvature. Given any Po E M, let fpo (x, t)

be defined by H (x,po, t) = (47rt)-n/^2 e-fpo(x,t) being the heat kernel (min-
imal positive fundamental solution). Recall that Wlin (jp 0 ( t) , t) is well de-
fined and finite (see Corollary 16.17 in Part II). Then the following two
properties are equivalent:

(1) (g has maximum volume growth) There exists c > 0 such that

VolB (p, r)

----2'.c rn

for all p EM and r E (0, oo).

(2) (Linear entropies of heat kernels are uniformly bounded) There

exists C < oo such that for all po E M

IWun (fp 0 (t), t)I:::::; C for all t E (0, oo).

(^11) Needless to say, where C is independent of (po, ta).
(^12) At the time of this writing we are not aware of a complete proof of this claim in
the literature.
(^13) Some of this material was also discussed in Sections 1-3 of Chapter 16 in Part IL