1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

26 17. ENTROPY, μ-INVARIANT, AND FINITE TIME SINGULARITIES

LEMMA 17.25. Let (Mn,g) be a closed Riemannian manifold. The quan-

tity JM w^2 log ( w^2 ) dμ depends continuously on w with respect to the £^2 (l+<5) _

norm for any 8 > O. In particular, the dependence is continuous with respect

to the W^1 >^2 -norm.

PROOF. To see this, suppose w1, w2 E £^2 (l+<5) (M), where 8 > 0. At

each x E M we have

w~ log ( w~) -wi log ( wi)

1

lw2I d

= -d ( u^2 log ( u^2 )) du

lw1I u

1

lw2I

= 2u ( 1 + log ( u^2 )) du.

lw1I

Applying the mean value theorem for integrals to this, we have

w~ log (w~) -wi log (wi) = (lw2l - lw1I) · 2a (1 +log (a^2 )),

(17.79)

(17.80) la log al:::; max { ~'

5

1

e a1+^8 }

since for a, 8 > 0 we have a log a 2: -~ and log a :::; Je a^8 • Therefore

we can bound JM 4a^2 (1 +log ( a^2 ) )

2

dμ in terms of llwillL2(1+o)(M,g) and

llw2llL2(1+o)(M,g)· The lemma now follows from (17.79). D

3.2. Strong maximum principle for weak solutions.

We now give the proof of the strong maximum principle for weak so-

lutions, 9 which is used in the proofs of Proposition 17.20 and Proposition

17.24. The following proof is on pp. 114-116 of Rothaus [161] (we also

fallows his notation for the most part).

LEMMA 17.26 (Strong maximum principle for weak solutions). Suppose

(Mn, g) is a complete Riemannian manifold. Let w 00 2: 0 be a 01 ,a function

which is a weak solution to (17. 77). If w 00 (p) = 0 for some p E M, then

w 00 = 0 in a neighborhood of p.

(^9) 0n the other hand, Calabi [21] proved strong maximum principles for sub-and super-
solutions in the support sense; see also Trudinger (181] and Theorem 2.4 of Andersson,
Galloway, and Howard [5].