1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

48 1. RICCI SOLITONS

LEMMA 1.79. The scalar curvature o{t>g and ttg are given by

b R = R-q.6.logA-q (q: l) IVlogAl^2 ,

ttR=R+q.6.logA-q(q:l) IVlogAl^2.

EXERCISE 1.80. Show that if A~ exp ( -~f) (i.e., log A= -~f), then

b 1

Rij = Rij + ViVjf--VdVjf,

q

b R = R + 2.6.f - q +

1

. q IV f^12 '.

tt R = R - 2.6.f - q +

1

IV f 12.

q

Hence

q-->oo lim Rb = R + 2.6.f - IV fl^2 ,

lim b Rij = Rij + ViVjf

q-->oo.

are Perelman's modified Ricci tensor and scalar curvature (see also (5.18)

and (5.19)).

As a consequence of the above exercise, if h has unit volume, we then

have

JMxP R (bg) dμo 9 =JM ( R+ 2.6.f - q;

1

1Vfl

2

) e-f dμ 9

=JM ( R+ q~

1

1Vf1

2

) e-fdμ 9.

Taking q---+ oo, this limits to :F (g, f) defined in (1.68).

9.2. Buscher duality. For a warped product solution of the modified

Ricci flow of the above type we have the following.

THEOREM 1.81 (Buscher duality). If

b g (t) = g (t) +exp(-~ f (t)) t dy°' ® dy°',

q a=l

t E I, satisfy the modified Ricci flow

(1.81) it^0 9ab = -2 (^0 Rab+ 2 °V a^0 Vb^0 ¢) ,

where b ¢ is a function on M x I, then the dual metrics

Hg (t) = g (t) +exp Gi (t)) t, dy" 0 dy"