1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

454 A. BASIC RICCI FLOW THEORY

so that Cut (p) = expP (8Cp). The cut locus is a closed set with measure

zero. We have

expPjintCp: Cp\8Cp-+ M\ Cut (p)

is a diffeomorphism. Let 8 / 8r = 1 ; 1 I::~=l xi 8 ~i denote the unit radial vector

field on TpM - { 0}. If x rf:. Cut (p) U {p} , then r is smooth at x, V' r ( x) =

(exppt 8/8r, and IV'r (x)I = 1.

Suppose that we have a function F : M x [O, T) -+ JR which satisfies the

differential inequality

(A.15) (!! 8t - !:>,) F -< c2 - p2

for some constant C. For an example of such a function, see the proof of the

local first derivative of curvature estimate in Part II of this volume.

If M is a closed manifold, then we can apply the maximum principle to

F to obtain the estimate

F ( x, t) ::; C coth (Ct) ,

where the RHS is the solution to the ODE 1t = C^2 - j2 with limt\,,0 f (t) =

+oo. On the other hand, if Mis noncompact, then one way of obtaining an

estimate for F is to localize the equation by introducing a cut-off function.

In particular, suppose (Mn,g) satisfies Re 2: - (n-1) K for some K >

0. Let rJ : [O, oo) -+ JR be a smooth nonincreasing function satisfying rJ( s) = 1
   for 0 ::; s ::;! and rJ( s) = 0 for s 2: 1. We may assume

(A.16) 0 2: rJ^1 2: -607 and - Co.fi7::; r]^11 ::; Co,

where Co is a universal constant.^2 Given p EM and A> 0, define ¢(x) ~

rJ ( d(~p)). Recall that in M\ ( {p} U Cut (p))

IV' d (., p) I = 1, b,.d (., p) ::; ( n - 1) JK coth ( JK d (., p)) ,

where the Laplacian estimate follows from (A.9) and the assumption Re 2:

• ( n - 1) K. Hence at points x rf:. Cut (p) , ¢ is smooth and

(A .17 ) I V'</J 12 ::;C¢, b,.= ArJ^1 I .b,.d+ A^1 2 rJ II I \i'd^12 2:-Cy</J, ri.

where C depends on A (and where we used (A.16), rJ^1 ::; 0, and the fact that

the support of r]^1 ( d7)) is contained in B (p, A) \B (p, 4) ).

We calculate that for x rj:. Cut (p) ,