1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

276 6. ENTROPY AND NO LOCAL COLLAPSING

COROLLARY 6.89. If Zt9ij = -2~j and

~~ =~f+a(R+2~f-1Vf1

2

),

for some a E JR, then the energy on a closed manifold Mn satisfies

d:F (g (t) 'f (t)) = .!!_ r (R + 2~f - IV 112 ) e-f dμ

dt dt }M

= 2 JM IRij + ViVjfl^2 e-f dμ

-(l+a) JM (R+2~f-IVJl^2 )2 e-fdμ.

So if a::::; -1, then

d

d f (R+2~f-1Vfl^2 ) e-f dμ 2:: o.

t~.

EXERCISE 6.90. Put a positive constant E in the above formulas; more

precisely, consider the E-entropy We: and determine equations for f and T

such that a monotonicity formula for We: holds. Perhaps one should consider
the set of equations

0

ot9ij = -2Rij

of = ~f - ~ (f -1.!.) +a (R + 2~f - IV fl^2 - ~ (f - n))

{)t T 2 T

dT

-=E

dt

for a::::; -1.

7 .2. Second variation of energy and entropy. As is typical for

energy-type functionals, we consider the second variation of Perelman's en-

ergy and entropy functionals. We are particularly interested in what sense

critical points of the entropy functional are stable, i.e., have nonnegative

second variation. ·

Suppose

09

= v and

0

f = h

OS OS

on a closed manifold Mn. Equation (6.31) implies the following:

:s [(Rij + ViVjf) e-f dμ]

(6.107) = \7p (e-f :srfj) dμ

• [ ViVj ( h - ~) - (Rij + ViVjf) ( h - ~)] e-f d~.