1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

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

LEMMA 17.1 (Logarithmic Sobolev inequality). Let (Mn,g) be a closed

Riemannian manifold. For any a > 0, there exists a constant C (a, g) such

that if <p > 0 satisfies JM <p^2 dμg = 1, then

(17.16) JM <p^2 log<pdμg:::; a JM jV'<pj~dμg + C(a,g),

where^1

(17.17)

2

C ( a,g ) _ -aVo l ( g )-2/n + 4ae2Cs(M,g)" n

Here Cs (M,g) denotes the L^2 Sobolev constant, which we define to

be the best (largest) positive constant such that (see Lemma 2 in [114])

(17.18)

n-2

JM jV'<pj~dμg;::;,: Cs (M,g) (JM <p_;;::2dμg)--;;;- -Vol(g)-~ JM <p^2 dμg

for any C^00 function <p on M.

We have the following elementary properties.

LEMMA 17.2 (Sobolev constants under scaling the metric). Let (Mn,g)

be a closed Riemannian manifold.

(i) The L^2 Sobolev constant has the property that for any .A.> 0,

Cs (M, .A.^2 g) =Cs (M, g).

(ii) The logarithmic Sobolev constant has the property that for any a >

0 and .A. 2: 1,

C(a,.A.^2 g) :'.S:C(a,g).

PROOF. (i) Let g = .A.^2 g and (jJ = ,A-n/^2 <p. Then dμ9 = .A.ndμg, <p^2 dμ9 =

<p^2 dμg, and

JM 1v<p1; dμg = A-2 JM jV'<pj~ dμg

n-2

;::;,: .A.-^2 Cs (M,g) (JM <pn

2

::2dμg )----,;:---- .A.-^2 Vol (g)-~ JM <p^2 dμg

n-2

=Cs (M,g) (JM <pn

2

::2dμ9 )----,;:---- Vol(g)-~ JM (jJ^2 dμ9.

From this we can easily deduce that

(17.19) Cs (M, .A.^2 g) =Cs (M, g)'

i.e., the L^2 Sobolev constant is invariant under scaling the metric.

(^1) 1n (17.17) we correct formula (6.67) in Part I, where the n (^2) /4 factor originally
appeared in the denominator; after (6.66) it should read Cn = 2 ~.