1547671870-The_Ricci_Flow__Chow

2. LOCAL VERSUS GLOBAL GEOMETRY 301

Assume there is B E IR such that Re 2:: (n - 1) Bg. Wherever r is

a smooth generalized distance function, taking f = r in the Bochner-

Weitzenbock formula gives the estimate

0 = l\7\7rl^2 + (\7r, \7 (~r)) +Re (\7r, \7r)

(B.2) 2:: n ~

1 (~r)

(^2) + (\7r) (~r) + (n - 1) B,
since (\7r, \7 (~r)) = d (~r) (\i'r) = (\7r) (~r). If we define
1

v~ --~r,

n - 1

then (B.2) is equivalent to the inequality

(B.3) dv (r) = (\7r) (v)::::; - v^2 - B.

This suggests we compare v (r) with the solutions of the Riccati equation

namely

(B > 0)

(B = 0)

(B < 0)

dw 2

• =-w - B
  dr '

lim w = oo,

r-->0+

w (r) =VB cot ( VBr)

1

w (r) = -

r

w (r) = J=B coth ( J=Br).

We can indeed make this heuristic rigorous in the following case.

PROPOSITION B.39. Let (Mn,g) be a complete Riemannian manifold

such that Re 2:: (n - 1) Bg for some BE IR. If r (x) ~ d (p, x) is the metric

distance from some fixed p E Mn, then on a neighborhood of p contained

inMn \ ( {p} U Cut (p)), we have the estimate

~r

• 
  
  
  
  • <
    n - 1 -
  
  
  
  

JBcot ( JBr) if B > 0

l/r if B = 0

FE coth ( FBr) if B < 0.

PROOF. Note that r is smooth on Mn\ ( {p} U Cut (p)), and ~r satisfies

lim r~r = n - 1.

r-->0+

Thus the function defined by

u(x)"' {

n - 1

b,.r

0

if x i-p

if x = p