- CONVERGENCE AT ALL TIMES FROM CONVERGENCE AT ONE TIME 137
2.2. Convergence at all times from convergence at one time.
2.2.1. The Arzela-Ascoli theorem. With uniform derivative bounds on
the metrics in the sequence, the compactness theorem will follow from the
Arzela-Ascoli theorem.LEMMA 3.14 (Arzela-Ascoli). Let X be a <Y-compact, locally compact
Hausdorff space. If {fkhEN is an equicontinuous, pointwise bounded se-
quence of continuous functions fk : X --+ IR., then there exists a subse-
quence which converges uniformly on compact sets to a continuous function
Joo: X--+ R
The reader is reminded that <Y-compact simply means that the space
is a countable union of compact sets, and hence any complete Riemannian
manifold satisfies the assumption.
COROLLARY 3.15 (Metrics with bounded derivatives preconverge). Let
(Mn, g) be a Riemannian manifold and let K C Mn be compact. Further-
more, let p be a nonnegative integer. If {9khEN is a sequence of Riemannian
metrics on K such thatsup sup [V'°'gk[ :::; C < oo
o:::;a:::;p+l xEKand if there exists 8 > 0 such that gk (V, V) :2: 8g (V, V) for all V E TM,
then there exists a subsequence {gk} and a Riemannian metric g 00 on K
such that gk converges in GP to g 00 as k --+ oo.
PROOF (SKETCH). We need to show that {(9khJkEN form an equicon-
tinuous family. We use the fact that in a coordinate patch
V' a (gk)bc = a~a (gk)bc - r~b (gk)dc - r~c (gkhd.
Thus if [ V' 9k [ is bounded, then \ a~a (9k he\ is bounded for each a, b, c in
each coordinate patch. Hence, by the mean value theorem, the (gk)bc form
an equicontinuous family in the patch and there is a subsequence which
converges to (g 00 )bc. Since K is compact, we may take a finite covering by
coordinate patches and a subsequence which converges for each coordinate
patch. We have thus constructed a limit metric. Note that the uniform
upper and lower bounds on the metrics gk ensure that 900 is positive definite.
Similarly, we can use the bound on \ V'^2 9k \ to get bounds on second
derivatives of the metrics \ ax~~xd (gkhc\ in each coordinate patch and thus
show the first derivatives are an equicontinuous family. Taking a further
subsequence, we get convergence in C^1. Higher derivatives are similar. D
2.2.2. Proof of the compactness theorem for solutions assuming the com-
pactness theorem for metrics. We will now use Corollary 3.15 together with
Lemma 3.11 to find a subsequence which converges and complete the proof
of Theorem 3.10. Recall that we have assumed Theorem 3.9 and hence there