1547671870-The_Ricci_Flow__Chow

3. BUSEMANN FUNCTIONS

Hence by Lemma B.52,

Cs~ n IHI'Ys = n {x E Mn: b'Ys (x) :SO}

)'ER(O) 'YER(O)

= {x E Mn : b'Y (x) :S s for all"( ER (O)}

= {XE Mn : b (x) :S s}.

311

D

COROLLARY B.60. Ifs > 0, then Cs contains the closed ball of radius s

at 0.

PROOF. By Lemma B.48, we have b (x) :S d (0, x), and hence

Cs= {x E Mn: b(x) :S s} ~ {x E Mn: d(O,x) :S s} ~ B(O,s).

D
COROLLARY B.61. For any choice of origin 0 E Mn, one has Us20Cs =
Mn.

D
PROPOSITION B.62. For every choice of origin 0 E Mn ands E [O, oo),
the set Cs is compact and totally convex.
PROOF. By Proposition B.54, Cs is the intersection of closed and totally
convex sets, hence is itself closed and totally convex.
Suppose Cs is not compact. Then there exists a sequence of points Pi E

Cs with d(O,pi)-+ oo as i-+ oo. For each i, let /Ji: [O,d(O,pi)]-+ Mn be

a unit-speed minimal geodesic from 0 to Pi· Since Cs is totally convex, each

/3i C Cs. After passing to a subsequence, we may assume the unit tangent

vectors /Ji (0) converge to a unit vector V 00 E ToMn. Let (3 00 : [O, oo)-+ Mn

denote the geodesic with /3 00 (0) = V 00. As in Lemma B.42, we observe that

(3 00 is a ray such that the images /3i -+ (3 00 uniformly on compact sets. Now

for any t E (0, oo ), consider the point (3 00 (s + t). Then /3i (s + t) E Cs is

defined for all i large enough, and

i-+oo Em d (f3i(s + t), (3^00 (s + t)) = 0.

Since Cs is closed, this implies (3 00 (s + t) E Cs, which contradicts the fact

that (3 00 (s + t) E IIB(/3oo),. D

COROLLARY B.63. The Busemann function b associated to a point 0 E

Mn is bounded below.

PROOF. b (x) is continuous and Cs = { x E Mn : b (x) :S s} is compact.

D

We have seen that the totally convex sets Cs exhaust Mn. The next

result refines Lemma B.59 and gives the sense in which the level sets of b

are parallel.