1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

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

PROOF. By the scaling invariance ofμ (see property (iii) on p. 236 of

Part I) and (17.21), we have for any c E (0, oo),

1 n

μ(g,7) = μ(cg,c7):::; c7>.(cg) +; Vol(cg)-

2

1og(47rc7)-n.

Taking c =Vol (g)-^2 /n, we obtain

1 n

μ(g,7):::; c7>.(cg) + ;-

2

1og(47r7) +logVol(g)-n

and (17.22) follows from).. (cg)= c-^1 >. (g) ::=:; 0. D

As a consequence of Corollary 17.4 we have that if ).. (g) < 0, then

lim 7 _, 00 μ (g, 7) = -oo. This improves Exercise 6.32 in Part I.

1.2.2. Lower bounds for μ.

Now we consider lower bounds forμ using the logarithmic Sobolev in-

equality; we have the following.

LEMMA 17.5 (Lower bound forμ). Let (Mn,g) be a closed Riemannian

manifold and let 7 > 0. We have

n

(17.23) μ (g, 7) 2:: 7 Rmin (g) - 2C (27, g) - 2 log(47r7) - n,

where Rmin (g) ~ minxEMR ( x) and the constant C (27, g) is given by (17 .17).

PROOF. By Lemma 17.1, for any w 2:: 0 with JM w^2 dμg = 1 we have

JM w^2 logwdμ:::; 27 JM j\7wj^2 dμ + C (27,g).

Substituting this into (17.7), we obtain

K(g,w,7) =JM (7Rw^2 +47j\7wl^2 ) dμ-~log(47r7)-n

• 2 JM w^2 logwdμ
  n
  2:: 7 Rmin (g) - 2C (27, g) - 2 log(47r7) - n

since JM 7Rw^2 dμ 2:: 7Rmin (g). Taking the infimum over w, we obtain the

desired inequality (17.23). D

Let Cs (g) denote the L^2 Sobolev constant: the smallest number such

that

(17.24)

ll'PllL~(g) :::; Cs (g) ll'Pllw1,2cg) ~Cs (g) (JM (l\7cpj

2

+ cp

2

) dμ)

112

for all cp E C^00 (M). By (17.18), we have

Cs (g)

2

:::; Cs(~, g) max {Vol (g)-~, 1}.