1547671870-The_Ricci_Flow__Chow

82 3. SHORT TIME EXISTENCE

LEMMA 3.15. If {Xt: 0 :St< T :S oo} is a continuous time-dependent
family of vector fields on a compact manifold Mn, then there exists a one-
parameter family of diffeomorphisms {'Pt : Mn ---) Mn : 0 :S t < T :S oo} de -
fined on the same time interval such that

(3.37a)

(3.37b)

8<.pt

ot (x) = Xt ['Pt (x)]

<po (x) = x

for all x E Mn and t E [O, T).

PROOF. We may assume there is to E [O, T) such that <.p 8 (y) exists for

all 0 :S s :S to and y E Mn. Let ti E (to, T) be given. We shall show that 'Pt

exists for all t E [to, t 1 ]. Since t 1 is arbitrary, this implies the lemma. Given

any x 0 E Mn, choose local coordinate systems (U, x) and (V, y) such that

xo E U and 'Pto (xo) E V. As long as x E U and <pt (x) E V, the equation

(3.37a) is equivalent to

gt [yo 'Pt o x-1 (p)] = y* [ O~t [x - 1 (p)]]

= (y*Xt o y-^1 ) (yo 'Pt o x-^1 (p))

for p E x (U) such that 'Pt o x-^1 (p) E V. Setting Zt = y o 'Pt o x-^1 and

Ft= y*Xt o y-^1 , we get

f)

f)t Zt =Ft (zt)

where Zt and Ft are time-dependent maps between subsets of !Rn. Thus we

see that (3.37a) is lo cally equivalent to a nonlinear ODE in !Rn. Hence for all

x EU such that 'Pto (x) E V , a unique solution to (3.37a) exists for a short

time t E [to, to+ c). Since the vector fields Xt are uniformly bounded on the

compact set Mn x [to, t1], there exists an t > 0 independent of x E Mn and

t E [to, t1] such that a unique solution 'Pt (x) exists fort E [to, to+ t]. Since

the same claim holds for the flow starting at 'Pt+£ ( x), a simple iteration

finishes the argument. D

REMARK 3.16. The lemma may fail to be true if Mn is not compact.

For example, if M^1 =IR and Xt (u) = u^2 , then we have the ODE

f) 2

at r.pt( x) = ['Pt ( x) J

<po (x) = x

whose solution is 'Pt (0) = 0 if x = 0 and

1

'Pt (x) = x-l - t

if x -=/= 0. If x < 0, the maximal solution exists fort E ( x -^1 , oo). If x > 0, the

maximal solution exists for t E (-oo, x-^1 ). In this case, we note that only

<po is a diffeomorphism of IR, whereas fort > 0 we have the diffeomorphism

'Pt : (-oo, c^1 ) ---) (- c^1 , oo).