1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

226 6. ENTROPY AND NO LOCAL COLLAPSING

THEOREM 6.4 (Entropy monotonicity for Ricci fl.ow). Let (g(t),J(t),r(t)),

t E [O, T], be a solution of the modified evolution equations (6.14), (6.15),

and (6.16). Then the first variation of W along this solution is given by the

following:

d

dt W(g(t), f(t), r(t))

(6.17) =JM 2r IRij + \li'Vjf -

2 ~gijl

2

· udμ

(6.18) = 2r JM IRij - t ( \li'Vju - \li:'Vju +

2 ~ ugij)^1

2

· udμ 2: 0.

EXERCISE 6.5. Show that the line of reasoning of deriving Theorem 6.4

from Lemma 6.3 is rigorous.

REMARK 6.6. The expression

(6.19)

is exactly the matrix Harnack quantity for a solution u of the backward

heat equation. In particular, this same expression for positive solutions

of the backward heat equation appeared in Hamilton's derivation of the

monotonicity formulae for the harmonic map heat fl.ow, mean curvature

fl.ow, as well as the Yang-Mills fl.ow; see [184].^5

Equation (6.17) is Perelman's entropy monotonicity formula and im-

plies that W(g(t), f (t), r(t)) is strictly increasing along a solution of the mod-

ified coupled fl.ow except when g(t) is a shrinking gradient Ricci soliton (since

r > 0 in (6.17)), where it must fl.ow along 'Vf and where W(g(t),f(t),r(t))

is constant. This monotonicity is also fundamental in understanding the lo-

cal geometry of the solution g ( t) to the Ricci fl.ow as we shall see in the proof

of the no local collapsing theorem. The function f allows us to localize and

the parameter r tells us at what distance scale to localize ( JT) ..

Since we do not give the details of how to transform between the systems

(6.10)-(6.12) and (6.14)-(6.16), we also compute (6.17) directly through the

following exercise.

EXERCISE 6.7 (Deriving dJi from ft). Use the equation for ft to derive
the formula for dJi.

SOLUTION TO EXERCISE 6.7. The effect of the extra term+~ in (6.11)

as compared to (5.30) is to add

-;:_JM (R+ l'Vfl

2

) e-fdμ

(^5) For an exposition of the matrix Harnack estimate asssociated to (6.19), see [104].