1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

4. VOLUME GROWTH OF SHRINKING GRADIENT R ICCI SOLITONS 23

together, we obtain the !-Bochner formula

(27.96)

1

2.6.1l'Vul^2 = l\7^2 u l^2 +('Vu, \7.6.1u) + Rc1(\7u, 'Vu).

Let 6 EM and let r(x) = d(x, O). Let S(O,r) ~ {x E M : d(x, O) = r} be

the distance sphere. Let H denote the mean curvature of S( 6 , r) , wherever it is a

smooth hypersurface. The f-mean curvature is

(27.97) H1 = H - (\i'f, \i'r) = .6.1r

since .6.r = H. Recall that since l'Vrl^2 = 1 and \7^2 r =II, (27.96) with u = r implies

the f-Riccati equation

(27.98) aH^1 (a a )^2 (a a ) H

2

• = -Rc1 - , - -IIII ~ -Rc1 - , - - --.
  8r 8r 8r 8r 8r n-1

Let J denote the J acobian of the exponential map and let J f ~ e-f J denote the

!-Jacobian. We have tr lnJ =Hand tr lnJ1 =HJ. Thus

(27.99)

If Rct 2: -~g for some c: E IR, then

a r

2

H

2

( a a ) ( a

2

• (r^2 H) < 2r H - ---r^2 Re - - < n - 1 + r^2 - f + -c:).
  8r - n - 1 8r ' 8r - 8r^2 2

Hence

(27.100) -a ( r^2 H t ) < n-1-2r-af + c: - r^2.

8r - 8r 2

Integrating this while using limr '\,O r^2 HJ (r) = 0, we obtain

(27.101)

H (r) - a -f (r) =Ht (r) ~ --n -^1 + - r c - -^2 1r -r-(a f r)dr- -.

8r r 6 r^2 0 8r

Now co nsider the case where c: = 0. Wherever we have the bound l\7 fl ~A,

(27.102)

n- 1

H t (r) ~ --+A.

r

By substituting this into (27.99), we have

Taking r 1 -+ 0 and calling r 2 = f , we obtain

(27.103)