1547671870-The_Ricci_Flow__Chow

7. HIGHER DERIVATIVE ESTIMATES AND LONG-TIME EXISTENCE 203

PROPOSITION 6.48. Let (Mn,g(t)) be a solution of the Ricci flow on a

compact manifold with a fixed background metric g and connection 'V'. If

there exists K > 0 such that

IRm (x, t)l 9 :SK for all x E Mn and t E [O, T),

then there exists for every m EN a constant Cm depending on m, n, K, T,
go, and the pair (g, 'V') such that

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

The first step in proving the proposition (and ultimately in establishing
long-time existence of the normalized flow) is to obtain a sufficient con di ti on
for the metrics composing a smooth one-parameter family to be uniformly
equivalent. Recall that one writes A :S B for symmetric 2-tensors A and B if
B - A is a nonnegative definite quadratic form, that is if (B - A) (V, V) 2: 0
for all vectors V.

LEMMA 6.49. Let Mn be a closed manifold. For 0 :S t < T :S oo, let
g (t) be a one-parameter family of metrics on Mn depending smoothly on
both space and time. If there exists a constant C < oo such that

frl~g(x,t)I dt:::;c

lo ut g(t)

for all x E Mn, then

e- c g (x, 0) :S g (x, t) :S e^0 g (x, 0)

for all x E Mn and t E [O, T). Furthermore, as t / T, the metrics g (t)

converge uniformly to a continuous metric g (T) such that for all x E Mn,

e- c g (x, 0) :S g (x, T) :S e^0 g (x, 0).

PROOF. Let x E Mn, to E [O, T), and V E TxMn be arbitrary. Then

using the fact that IA (U, U)I :S IAl 9 for any 2-tensor A and unit vector U,

we obtain

\log(:~::~:((~,~;) I= \ foto gt [logg(x,t) (V, V)] dt \

= ot (x,t) ' dt

1

to 2-g (V V)

o g(x,t) (V, V)

:S foto I gtg(x,t) ( l~I' l~I) I dt

:S rto I ~ g(x,t) \ dt :S c.

lo ut g(t)

The uniform bounds on g ( t) follow from exponentiation.