- VOLUME GROWTH OF SHRINKING GRADIENT RICCI SOLITONS 25
By combining (27.107) and (27.108), we obtain(27.110) ~ (nn J(f)) = ln J(f) + f~ ln J(f)
of fn-^1 fn-^1 of fn-^1
f^2 of
:::; -4 -f(f) + f(O) +for (f)(of f)
2
( (of)2
:::; f (O) - or (f) - 2 - f (f) - or (r) ).Let \l fr be the tangential component to oB 0 (r) of \l f. Then f = l\l fl^2 + R =
( U )^2 + l\l JTl^2 + R. By integrating (27.110) from f = 0 to f = r, we obtain(27.111) J(r)^11 r ((of f)
2
ln - )1 :::; f(O) - - ~ (f) - - + l\l frl
(^2) (f) + R(f) df
rn- r 0 ur 2
:::; f(O).
Therefore
(27.112)
Since J is the volume density of g in spherical coordinates, by integrating (27.112),
we obtain (27.106). D
EXERCISE 27.43. Show that
- ( J(f) )f
r H ef(O)fn-l
is nonincreasing and that the limit as f --+ 0 is equal to l.
REMARK 27.44. Suppose that the minimum of R is attained at 6 and that
l"Vfl (0) = 0. Then we have R(O) = f(O) = f (0). This implies that~ J; R(f)df;:::
R(O) = f (0). Therefore, by (27.111),
(27.113) ln J(r) n-l :::; f(O) - -^1 1 r R(f)df:::; 0.
r r 0This implies that VolB 0 (r) :::; wnrn. On the other hand, by Corollary 27.36, we
know that VolB 0 (r):::; C(Q, O)rn-2R(6).
EXERCISE 27.45. Show that_ efi-J(r)+~ J; f(r)dr J(f) e~ J;((;-J'(r))
2
+J(r)-(J'(r))
2
)dr J(f)
(27.114) r H fn-l = --------fn-l ------'--is nonincreasing and that the limit as f --+ 0 is equal to e f (O).
Let 6 EM, x tJ_ Cut(O), and r(x) = d(x, 0), where Cut(O) is the cut locus of
- Define
r2(x) 2 r<x)
h(x) = 12 - f(x) + r(x) Jo f (r(s)) ds,
where 'Y : [O, r(x)] --+ M is the unique minimal unit speed geodesic joining 6 to x.
The measure corresponding to the numerator in (27.114) is
dm (x) = eh(x)dμ (x) on M - Cut(O).