1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

5. PERELMAN'S FORMALISM IN POTENTIALLY INFINITE DIMENSIONS 427

(6)

by

g~ = ~:~ (x, y, T) °:rd (x, y, T) 9cd (cp (x, y, T))

tr^0

= ~ (x,y,T)§ao(x,y,7)

=0.

Now define the 1-parameter family of diffeomorphisms

'l/Jr:M-+M

0

OT '!/Jr= (\7 f)('l/Jr(x), T).

Consider the diffeomorphism

defined by

and the diffeomorphism

defined by

Let

g m __,_ --;--{l!*-m g '

which is a Riemannian metric on M.

We compute the components of the metric gm. First note that

2'!/J; (Hess 9 J) = 'l/J; (.Cgrad g tg) = £.1.*( 'P-r gradg f ) ('lj;;g)

= .Cgrad'ef;;.g(fow) ('lj;;g)

= 2Hess1/Jig (f o '1!).

D

LEMMA 8.42. Let (Mn, g ( T)) be a solution of the backward Ricci flow

and let f satisfy ( 6.15)

of= ~f-l\7fl^2 + R-~.

OT 2T

The (spatial) components gYJ satisfy the following:

!gYJ = ('1j;;(2Rc(g) + 2\7\lf))ij mod O(N~^1 )

= 2Rc(gmlMx{y}x{r})ij + 2\7i\7jf o '1! mod O(N-^1 ),