3. 2 -DIMENSIONAL ANCIENT SOLUTIONS WITH FINITE WIDTH 57
of a comparison principle for metrics more technical, we instead use complex ge-
ometry. R ecall the following complex 1-dimensional sp ecial case of a result of Yau
(see Theorem 2 in [444]).
LEMMA 28.46 (Generalization of the Schwarz- Ahlfors- Pick lemma). Let(Mr, gl) and ( M§, g2) be Riemannian surfaces such that 91 is a complete metric,
R9 1 2: -c1, and R9 2 :S -c2, where c1 2: 0 and c2 > 0. If if>: (M1,g1)---+ (M2, g2)
is a conformal map, then
(28.53)In particular, if 92 and g1 = vg2 are Riemannian metrics on a surface M^2 , where
g 1 is complete with R 91 2: -c1 and R9 2 :S -c2 for constants c1, c2 > 0, th en v 2: ~.
By the Schwarz lemma and the method of Giesen and Topping [120], one
obtains the following consequence of an estimate of Ana Rodriguez, Juan Luis
Vazquez, and Juan R. Esteban.LEMMA 28.4 7 (The hyperbolic cusp is a lower ba rrier). For each t < 0 there
exists c( t) > 0 such that
(28.54) u(s, e, t)::::: c(t)s-^2
for alls= ln r suffi ciently large and e E 51. The constants c(t) are bounded from
below by a positive constant on each compact subinterval of ( -oo, 0). Since u = r~,
this is equivalent to(28.55)
c(t)
u(r,B, t) 2: r2(lnr)2·PROOF. Let (JR.^2 ,g(t)), t E (-oo,O), be our solution. On JR.^2 - Bo(l) considerthe metric 9hyp = r 2 ( 1 ; 1 r) 2 9IE· This is the complete hyperbolic cusp as can be seen
from the change of variables CJ= ln (ln r), which yields
9hyp = dCJ^2 + e-^2 " dB^2 for - oo < CJ < oo, B E 51.
Let T/ : JR.^2 ---+ [O, l] be a smooth radial cutoff function with T/ = 1 in Bo(2) and
T/ = O in JR.^2 - B 0 (3). On JR.^2 - Bo(l) define the "blended" complete metric
9b1e(t) = (r ln r)-^2 '7 u(t)^1 -?JgJE.
WehaveR9bie (r, B, t) = -2forr E (1,2] andR 9 bie (r,B, t) = Rg(t) > Oforr E [3,oo).
Since the blended region is compact, there exists a constant C(t) < oo dep ending
only on T/ and g(t) such that
R 9 ble (r, e, t) 2: -C(t) in (Bo(3) - Bo(2)) x (-oo, 0).
Hence R9b 1 • ;:::: -C(t) in (JR.^2 - Bo(l)) x (-oo, 0), where C(t) is uniformly bounded
on compact subintervals of (-oo, 0).
Now, by applying Lemma 28 .46 to 91 = 9ble(t) and 92 = 9hyp with if>= id , we
obtain
(r ln r)^2 (l-?J) u(t)^1 -?J > -
2
- in JR.^2 - Bo(l).
- C(t)
In particular, we have
2 2 2