1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

274 6. ENTROPY AND NO LOCAL COLLAPSING

is a bounded nonnegative solution to the Lichnerowicz Laplacian heat equa-

tion gtv = t:J..Lv, where t:J..L is defined by (Vl-3.6), then under measure-

preserving variations (v (t), h (t)), where 6(v(t),h(t)) (e-f(t)dμg(t)) = 0, we

have

6(v(t),h(t)) ( R + 2/::J..f - l\7 fl^2 ) (t) 2 0.

This follows from the linear trace Harnack estimate for ancient solutions.

EXERCISE 6.84. Show that extending (6.105) to nonmeasure-preserving
variations, we have the following. If 69 = v and <Sf= h, then

6(v,h) ( R + 2/::J..f - l\7 fl

2

)

(6.106) =div (divv) + (v,Rc) ~ 2 (divv) · \7f + Vij\7if\7jf

+2(/::J..-\7f·\7) (h-~)-2vij(Rij+\7i\7jf).

Note that (6.105) generalizes to the following, where \7 f is replaced by

a vector field X.

LEMMA 6.85. If 6g = v and 6X = \i'r, then

<5(v, vt) ( R + 2gij\7iXj -1x1^2 ) = \7i\7jvij + VijRij - 2\7ivikxk + VijXiXj

• Vij (2Rij + \JiXj + \7jXi).

PROOF. This follows from (Vl-p. 69d):

(^6) v R -- -t:J..v + \7i\7jv· iJ · - v· iJ ·R-iJ, ·
6( v,:r-vv) (-IXl^2 ) = v· iJ ·X·X i J · - X · \7v ,
and
6(v,vt) (2gij\7iXj) = 2\7i (:sxi)-2 (: 8 9ij) \7iXj -2lj (: 8 I'fj) Xk
= f:J..v - 2Vij\7iXj - 2\7iVikxk + \7kvXk.
D
EXERCISE 6.86 (Generalizing Exercise 6.84). Show that if 6g = v and
6X = Y, then Lemma 6.85 generalizes to
6(v,Y) ( R + 2gij\7iXj - IXl^2 ) =div (divv) + v ·Re -2 div (v) · X + v (X, X)

• Vij (2~j + \7iXj + \7jXi)

• 2 div ( Y -~ \7V) -2X · ( Y -~ \7V).

A different calculation yields