1547671870-The_Ricci_Flow__Chow

306 B. SOME RESULTS IN COMPARISON GEOMETRY

PROOF. If not, there exists E > 0, a sequence of points Pi E Mn with

d (pi, 0) / oo, and minimal geodesic segments cri joining 0 and Pi such that

Lo (0-i (0) ,p(O)) 2'. c

for each i and all rays p emanating from 0. By compactness of the unit
sphere in ToMn, there exists a subsequence such that limi--+oo 0-i (0) ~ V

exists. Let cr 00 : [O, oo) --+ Mn be the unique geodesic with cr 00 (0) = 0

and 0- 00 (0) = V. Arguing as in Lemma B.42, we see that cr 00 is a ray. In

particular, the condition

Lo (0-i (0), 0-(0)) 2'. c

is impossible. D

Recall that we already have a good upper bound for the Busemann

function associated to a point:

b (x) :::; d (x, 0).

The previous lemma lets us construct a lower bound for b in the event that

Mn has nonnegative sectional curvature.

COROLLARY B.50. If (Mn,g) is a complete noncompact Riemannian

manifold of nonnegative sectional curvature, then

b (x) 2'. d (x, 0) (1 - e (d (x, 0))).

PROOF. Given any point x E Mn, let a be a minimal geodesic segment

joining 0 and x. By the lemma, there exists a ray r emanating from 0 such

that

L ('Y (O), a (O)) :::; e (d (x, 0)).

Set y ~ / (d (x, 0)). By the Toponogov comparison theorem applied to the

hinge with vertex 0 and sides a joining 0 to x and 11 [O,d(x,O)J joining 0 to

y, we have

d(x,y):::; B(d(x,O)) ·d(x,O),

where the right-hand side is the length of an arc of angle e (d (x, 0)) in a

circle in the Euclidean plane of radius d (x, 0). Thus since b"I (y) = d (x, 0),

we get

b (x) 2'. b"I (x) 2: b"I (y) - d (x, y) 2: d (x , 0) [1 - e (d (x, O))],

where we used the fact that b"I has Lipschitz constant 1 to obtain the second

inequality. D

The gist of the this is that the Busemann function b associated to a point

0 is similar to the distance function to 0. However, from some points of

view, b has better properties. The property we shall be most interested in

is convexity.