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3.4. Vanishing of the first boundary term in the limit.
First, we show that (M^2 , g(t)) is asymptotically cylindrical at spatial infinity
in the following sequential sense.LEMMA 28.43 (Asymptotically cylindrical). Assume the hypothesis of Theorem
28.41. Given any sequence of basepoints Pi -t oo, the sequence of pointed solutions
(M,g(t) ,pi), t E (-oo,O), subconverges in the c= Cheeger-Gromov sense to a
static fiat cylinder.PROOF. By (28.46) we have that R is uniformly bounded and positive on M x(- oo, w] for each w < 0. Hence, by Shi's local first derivative estimate (see Theorem
14.10 in Part II) there exists a constant C < oo depending on w such that
(28.48) IVRI:::; C on M x (-oo,w].Claim. For each t < 0,
(28.49) lim R(p, t) = 0.
p-t=
Proof of the claim. By a result of Cheeger and Gromoll (see Theorem B.65 inVolume One), we have p ~ inj (9(t)) > 0. By this and the volume comparison for
R :::; C, there exists c' > 0 such that Area( Bq (p, t)) 2: c' for each q E M, where
Bq(p, t) denotes the geodesic ball of radius p centered at q with respect to g(t).
Now suppose that (28.49) is false. Then, by using (28.48), we see that there
exist c > 0 and a sequence of points Qi -t oo such that the balls Bq, (p, t) are
pairwise disjoint and f Bq,(p,t) Rdμ(t) 2: c for each i. This contradicts the Cohn-
Vossen inequality JM Rdμ(t) :::; 47r and so the claim is proved.
Now, by Hamilton's Cheeger- Gromov compactness theorem, for any sequence
of basepoints Pi -t oo, the sequence of pointed solutions (M, 9 (t), Pi), t E (-oo, 0),
subconverges in c= to a complete noncompact pointed solution (M~, 9=,P=),
t E (-oo, 0), with R 900 2: 0. From (28.49), we have R 900 (p=, t) = 0. By the strong
maximum principle, we conclude that g=(t) = 9= is fiat and static.
By Lemma 28.40 we have that (M=,9=) cannot be isometric to the Euclidean
plane. Since (M=, 9=) is orientable, fiat, complete, and noncompact, by the clas-
sification of such Riemannian surfaces we then have that (M=, 9=) is isometric to
a fiat cylinder. D
Next, infer from limp-t= R(p, t) = 0 that at each time the higher covariant
derivatives of R decay to zero at spatial infinity.
LEMMA 28.44 (Higher derivatives of curvature decay). For each t E (-oo, 0)
and j 2: 1 we have(28.50) lim IVj RI (p, t) = o.
p-t=PROOF. Given t < 0 and 0 EM, by (28.49) we have that for any c > 0 there
exists C 0 < oo such that
0 < R(p,t):::; c for p EM -Bo(C",t).
Thus the trace Harnack estimate (28.46) implies that
O<R(p,l)s;c forpEM-Bo(C 0 ,t) and tE(-oo,t].