1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

6. Notes and commentary

LEMMA 7.130 (Integration by parts inequality for£). We have

(7.148) JM (il£ - l\7£1^2 ) e-£dμ ~ 0.

PROOF. Since integration by parts holds for Lipschitz functions, we can

write Proposition 7.128(iii) as

(7.149) - JM \7£ (·, r) · \7cpdμ ~JM cpil£ (·, r) dμ,

where cp E C~ (M) is nonnegative. By an approximation argument as dis-

cussed above, this inequality holds for any nonnegative locally Lipschitz

function cp satisfying cp (q, r), l\7cp (q, r)I ~ c-^1 e -cd~(o)(p,q) for some constant

c > 0. Taking cp = e-£, we get

JM l\7£l^2 e-£dμ ~JM e-£il£dμ.

D

As a simple consequence of the above lemma and g~ -il£ + I \7£1^2 - R +

~ ~ 0 a.e., we obtain the following.

COROLLARY 7.131. For a solution to the backward Ricci flow (Mn, g.(r)),
r E [O, T] , with bounded sectional curvature, we have

(7.150) { (

8

£ - R + .'!!___) (4nrtl^2 e-edμ ~ 0

JM 8r 2r

and, for 0 < r1 < r2 < T,

(7.151)

1

72

1 ( ~8£ n ) /2

8

• R + - (4nrt e-£dμdr ~ 0.
  T1 M r 2r

10. Notes and commentary

1. This chapter is a discussion of §7.1 and §7.2 of Perelman's [297].


2. The reader may also consult Rugang Ye's notes on the £-function
   [382], which we have partially used as a source. See also the appendix of
   [134] for a brief discussion of reduced distance.


3. Notational conventions. In this chapter we have endeavored to
   maintain a consistent convention for the notation we have used. In particular
   we have used the following notation when discussing solutions to the forward
   and backward Ricci flow:
   ( M_n, g) : a static Riemannian manifold,

(Nn, h (t)): an arbitrary (not necessarily complete or with bounded

curvature) solution to the Ricci flow,

(Nn, h (r)), r E (A, 0): an arbitrary solution to the backward Ricci

flow,