1547671870-The_Ricci_Flow__Chow

304 B. SOME RESULTS IN COMPARISON GEOMETRY

LEMMA B.44. The functions b"f,s are uniformly bounded: for all x E Mn

ands 2: 0,

lb"f,s (x)I ~ d (x, 0).

The functions b"f,s are uniformly Lipschitz with Lipschitz constant l: for all

x, y E Mn ands 2: 0,

lb"f,S (x) - b"(,S (y)I ~ d (x, y).

And the functions b"f,s (x) are monotone increasing ins for each x E Mn:

ifs< t, then

PROOF. All three statements are consequences of the triangle inequality.

The first is immediate, and the second follows from observation

lb"f,s (x) - b"f,s (y)I = Id (y, I (s)) - d (x, I (s))I ~ d (x, y).

To prove the third, we note that db (s), I (t)) = t - sand observe that

b"f,t (x) ~ t - d (x, 1 (t))

2: s + (t - s) - d (x, 1 ( s)) - db (s) , I (t))

= s - d ( x, I ( s)) ~ b'Y ,s ( x).

D

The monotonicity of the pre-Busemann functions in the parameter s

enables us to make the following

DEFINITION B.45. The Busemann function b'Y : Mn___, JR. associated

to the ray I is

Since the family {b'Y,s} is uniformly Lipschitz and uniformly bounded

above, we can immediately make the following observations.

LEMMA B.46. The Busemann function b'Y associated to a ray 1 emanat-

ing from 0 E Mn is bounded above: for all x E Mn,

(B.5) I b'Y ( x) I ~ d ( x, o).

And b'Y is uniformly Lipschitz with Lipschitz constant l: for all x, y E Mn,

(B.6) lb'Y (x) - b'Y (y)I ~ d (x, y).

Intuitively, b'Y (x) measures how far out toward infinity x is in the direc-

tion of I · One could also regard b'Y as a renormalized distance function from

what one might think of as the 'point' r (oo). For example, in Euclidean

space, the Busemann functions are the affine projections. In particular, if