1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

230 6. ENTROPY AND NO LOCAL COLLAPSING

where u is defined in (6.3). When c: < 0, this is Perelman's entropy; when

c: = 0, this is Perelman's energy; and whens> 0, this is called the expander

entropy.^8 The definition of Ve is motivated by the following exercise.

EXERCISE 6.10 (Harnack Ve as an integrand for W 6 ). Let (Mn,g (t),f (t))
be a gradient Ricci soliton in canonical form. By Proposition 1. 7, the pair

(g (t), f (t)) satisfies

(6.27) Sfj ~ Rij + \l/\ljf +

2

:9ij = 0,

where s E R Note that if s 'I- 0, then g (t) and f (t) are defined for all t

such that T (t) > 0, whereas if c: = 0, then g (t) and f (t) are defined for all

t E (-oo, oo). Show that Ve (g (t), f (t), T (t)) is constant in space.

EXERCISE 6.11 (W 6 and the Gaussian soliton). Consider the Gaussian

soliton (ffi.n,gm;,f 6 ), where gm;= L:~=l (dxi)^2 and

Check that

(

_ 1~~

2

fort> 0 ifs< 0,

fc (x, t) ~ 0 fort E ffi. if c: = 0,

lxl

2

-« for t <^0 if E > 0.

a^2 f 6 E

(^8) xi. (^8) xJ. + -2 T 6ij = 0

for all t such that T (t) = st > 0, and thus Sfj = 0. Show Ve = 0, so

that W 6 = 0. It is useful to keep this example in mind, which reflects the
Euclidean heat kernel, when considering the function theory aspects of the
material in this chapter and especially the chapter on Perelman's differential
Harnack estimate in the second part of this volume.

EXERCISE 6.12 (Basic properties of W 6 ). Show that on a closed Rie-

mannian manifold (M.n, g) ,

(1)

We(§, f, T) = (47r)-n/^2 JM ( R9 + J\l fl~ -c(J - n)) e-f dμ9,

where g ~ T-^1 9,

( 2) for any constant c > 0

Wc(cg,f,cT) = WE:(g,f,T),

and

( 3) for any diffeomorphism <p : M ---+ M,

WC (<p*g, f 0 cp, T) = WE:(g, f, T).

(^8) These identifications are true up to constant factors.