484 I. ASYMPTOTIC CONES AND SHARAFUTDINOV .RETRACTION
(see (1.139) of [45] for example). Since the sectional curvature of (M,g)
is bounded above by ""^2 > 0, we have - (Rm (ax,·)·, ax) ~ -/'i,^2 I, so that
along ax we have the differential inequality
(I.45) V' ax Ilr ~ -/'i,^2 Ir -n;,
where Ir denotes 'the first fundamental form of 1-lr.
Given r ~ 0, suppose that Vr E Tax(r)1-lr is a unit vector and let V (r) E
Tax(r) 1-lr denote the parallel translation of Vr along ax.^15 We have IV (r) I =
1 and V (r) = Vr· If Vr is an eigenvector of Ilp, then (I.45) implies
·! lr=f (IIr (V (r), V (r))) ~ -i'i,^2 - (IIr (V (r), V (r)))
2
.In particular, if we define
,\ (r) ~ min IIr (W, W),
[W[=lthen, by applying.Lemma 10.29 in Part II to -,\,we have along ax,
~; lr=r ,\ (r) =min {! lr=f (IIr (V (r), V (r))): IIr (Vr, Vr) = ,\ (r)}
~min {-""^2 - (IIr (Vr, Vr))^2 : IIr (Vr, Vr) = ,\ (r)}
= ""2 ,\ (r)2.
Here we have used the elementary fact that if IIr (Vr, Vr) = ,\ (r), then Vr is
an eigenvector of IIr.
Furthermore, by (I.44) we have,\ (0) ~ -A, where A~ 0. Hence, by the
Sturmian comparison theorem for differential inequalities (see Angenent [6]
for example), we conclude that^16
IIr ~ ,\ (r) Ir~ /'i,COt (""r + cot-l (-/'i,-l A)) Ir
as long as the RHS is finite (i.e.,> -oo). (Note that cot-^1 (-/'i,-l A) E rn, ?r).)
Thus there are no focal poin~s of distance less than
(I.46) /,(A, /'i,) ~ /'i,-l (?r - C?t-^1 (-/'i,-l A)) > 0to 1-l .. This implies that
explN~(A,,.)1-l : N~A,i,;)1-l --+ N;,lA,i,;) (1-l)(^15) Note that parallel translation is an isometry between TO/,,,(fi) 1l and TO/,,,(r) Hr.
(^16) Note that the function ¢ (r) = A; cot ( A;r + cot- (^1) (-A;- (^1) 0)) is a solution to the ODE
def>= -A;2 - q},
dr
¢(0) = -C.
We leave it as an exercise for the reader to verify that the Sturmian comparison theorem
holds for differential inequalities using the lim inf of forward difference quotients. Actually
this last fact is used for the conventional 'PDE to ODE formulatfon' of the scalar maximum
principle.