1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

132 3. THE COMPACTNESS THEOREM FOR RICCI FLOW

2. Convergence at all times from convergence at one time

In this section we give the proof that the compactness theorem for Ricci
fl.ow (Theorem 3.10) follows from the compactness theorem at time t = 0
(Theorem 3.9). This is done by showing that bounds on the metric and
covariant/time-derivatives of the metric at time t = 0 extend to bounds on
the metric and covariant derivatives of the metric at subsequent times in the
presence of bounds on the curvature and covariant derivatives of curvature
(for all time). This is shown in subsection 2.1 below. The Arzela-Ascoli
theorem is then used to show that these bounds on the covariant/time-
derivatives of the metric imply that a subsequence converges to a solution
of the Ricci flow for all times (in subsection 2.2.2 below).

2.1. Uniform derivative of metric bounds for all time. In or-

der to extend the convergence at one time to convergence at all times, the
following derivative bounds need to be shown.

LEMMA 3.11 (Derivative of metric bounds at one time to all times).

Let Mn be a Riemannian manifold with a background metric g, let K be a

compact subset of M, and let gk be a collection of solutions to the Ricci flow

defined on neighborhoods of K x [,B, Vi], where to E [,B, Vil. Suppose that

(3.2)

(1) the metrics gk (to) are all uniformly equivalent tog on K, i.e., for

all V E TxM, k, and x E K,

c-^1 g (V, V) .-:::; gk (to) (V, V) .-:::; Cg (V, V)'

where C < oo is a constant independent of V, k, and x; and

(2) the covariant derivatives of the metrics gk (to) with respect to the

metric g are all uniformly bounded on K, so that

for all k and p 2: 1, where Gp < oo is a sequence of constants

independent of k; and

(3) the covariant derivatives of the curvature tensors Rmk (t) of the

metrics 9k ( t) are uniformly bounded with respect to the metric 9k ( t)

on K x [,B, Vi]:

/V1 Rmk/k .-:::; C~

for all k and p 2: 0, where C~ is a sequence of constants independent

of k.

Then the metrics 9k (t) are uniformly equivalent tog on K x [,B, Vi], e.g.,

(3.3) B (t, to)-^1 g(V, V) .-:::; 9k (t) (V, V) .-:::; B (t, to) g(V, V),

where

B (t t ) = C e^2 vn=TCb It-to I

' 0 '