1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

46 1. RICCI SOLITONS

THEOREM 1.75 (Expander entropy monotonicity formula). For a solu-

tion (g (t), f+ (t), r(t)), r (t) > 0, of (1.75)-(1.77) on a closed manifold

Mn,

d

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

=JM 2r \Rij + \h\ljf + + ;; \

2

(47rr)-nl^2 e-f+dμ 2:: 0.

Here -it W+(g(t), f+(t), r(t)) = 0 if and only if g (t) is an expanding

gradient Ricci soliton.

Recall that from Proposition 1. 7

(1.78) sij^6 ::::;= • Rij + \li\lj f + E G

27 9ij = o,

where r (t) ~ Et+l > 0 and~ denotes an equality which holds for a gradient

Ricci soliton (Mn, g ( t) , f ( t) , E) in canonical form.

EXERCISE 1.76. Show that

(1)

(1.79)

(2)

(1.80)

81t ~ -b.f-R+ l\lfl

2

• ~~'

Moreover, if f has a critical point in space, then C (t) is independent

oft.

9. Buscher duality transformation of warped product solit.ons.

In this section we describe an interesting duality transformation for solu-

tions of the modified Ricci fl.ow on certain warped products which in particu-

lar take gradient Ricci solitons to gradient Ricci solitons. The consideration

of these warped products with tori of potentially infinite dimensions leads

to Perelman's energy functional.

9.1. A metric duality transformation. Let (Mn,g) be a Riemann-

ian manifold and let (Pq, h) be a fl.at manifold such as a torus or Eu-

clidean space. Given a function A : M --* (0, oo) , consider the warped

product manifold (M, g) XA (P, h) which is the Riemannian manifold

(M x P, g +Ah). The Buscher duality transformation (see [38], [39])

takes the metric

to the metric

tt g ==. g +A-1h.

(This is a special case of T-duality in string theory.) Although its definition

is simple, the transformation has some surprising properties. Let {xi} ~=l