1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

28 27. NONCOMPACT GRADIENT RICCI SOLITONS

Therefore, by (27.125) we have

:tH(~(t)) :'.'.'. :tl(~(t)).
Under suitable growth conditions on the function ~' it can be shown that
limt--+oo ~ (t) = 0 and thus limt--+oo H (~ (t)) = 0. We conclude that if ~(t) satis-
fies (27.119), then
(27.128)
H (~(t)) :SI(~ (t)); that is, JM IV'~(t)l
2
eWl-f dμ :'.'.'.JM ~(t)ef.(t)-f dμ.

Given <p with JM e-'Pdμ = (4nt^12 , take ~(O) = f - <p. Then (27.128) at t = 0
implies that

JM (f - <p) e -cp dμ :S JM (IV' fl

2

+ IV'<pl

2

- 2 (\7 f , V'<p) )e-cp dμ

and

JM (IV'<pl

2

+ <p)e-'Pdμ :'.'.'. JM ( 2!~.f - IV' 11

2

+ f) e-'Pdμ.

We conclude from this and (27.118) that

(27.129)

JM (R + IV'<pl

2

+ <p - n)e-cpdμ :'.'.'. JM ( R + 26.f - IV' fl

2

+ f -n) e-cpdμ

= - (4nt^12 C1(9).

This finishes our sketch of the proof of Theorem 27.46.

In terms of w = e -cp/^2 , the logarithmic Sobolev inequality says that if JM w^2 dμ

= ( 4n t^12 , then

JM (Rw^2 + 4 IV'wl

2

• w

2

ln(w

2

))dμ :'.'.'. (n - C1(9)) JM w

2

dμ.

If we do not impose any constraint on w, then this is equivalent to

(27.130) JM w

2

ln(w

2

)dμ - ln (JM w

2

dμ) JM w

2

dμ :S JM (4 IV'wl

2

+ Rw

2

)dμ

• C2 (9) JM w

2

dμ,

where C2(9) = C 1 (9) - n - ln((4nt^12 ). Since R = ~ - 6.f, we obtain

(27.131)

JM w

2

ln(w

2

)dμ-ln (JM w

2

dμ) JM w

2

dμ :S JM (4 IV'wl

2

+('VJ, \7(w

2

)))dμ

+ C3(Q) JM w

2

dμ,

where C3(Q) = C 1 (Q) - ~ - ln((4nt^12 ).

REMARK 27.48. Compare the derivation of inequality (27.125) with Bakry and
Emery's proof of their logarithmic Sobolev inequality [16] (for an exposition, see
Proposition 5.40 in Part I).