1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

1. SHARAFUTDINOV RETRACTION THEOREM 461

1.2.· Manifolds with nonnegative sectional curvature.

When complete noncompact manifolds have nonnegative sectional cur-

vature, the notions introduced in the last subsection have the following prop-

erties.

1.2.1. Busemann functions.

We begin with Busemann functions (see for example Exercises 1.167 and

1.161 in [45]).

PROPOSITION I.12. Let r : [0, oo) -+ Mn be a ray.

(i) If (M, g) has nonnegative sectional curvature, then the Busemann

function b'l' is convex. Hence, for any p EM, bp. is convex.

(ii) If Rc 9 2". 0, then the Busemann function b'l' is subharmonic in the

sense of distributions.^2

As a consequence of Lemma I.4 we have that the Busemann function is
'almost' bounded below by the distance function (for a proof, see Corollary
B.50 of.Volume One).

COROLLARY I.13 (bp is almost bounded below by distance). If (Mn, g)

is a complete noncompact Riemannian manifold with nonnegative sectional

curvature and p E M, then for all x E M,

bp (x) 2". d (x,p) (1-(} (d (x,p))).

In particular, the .Busemann function bp is bounded below (see also
Corollary B.63 of Volume One). On the other hand, the Busemann function
b'l' associated to a ray is not, in general, bounded from below.

EXERCISE I.14. Give an example of a complete noncompact Riemannian

manifold (Mn, g) with positive sectional curvature and a ray r in M such

that b'Y is not bounded from below.

1.2.2. Half-spaces.
Next we discuss half-spaces in complete noncompact manifolds with non-
negative sectional curvature. By Lemma I.6, the next proposition follows
from Proposition I.12; see also Theorem 8.2 of [30].

PROPOSITION I.15 (sect 2". 0 - left half-spaces are totally convex). Let
(Mn, g) be a complete noncompact Riemannian manifold with nonnegative

sectional curvature. If r is a ray, then the closed left half-space lHI'Y is totally

convex.

Sublevel sets of the Busemann function have the following nice prop-
erty on manifolds with nonnegative sectional curvature. This follows from
Proposition I.15 and Corollary I.13 (see also Proposition B.62 of Volume
One).

(^2) See, for example, p. 373 in Part I for a definition.