- VOLUME GROWTH OF SHRINKING GRADIENT RICCI SOLITONS 19
(here we recall that by (27.46), f is a proper function). Hence we verify by the
co-area formula:
LEMMA 27.31. If e is a regular value off, then
V' (e) =;; l;fl dO',
{f=c}
(27.74)
R' (e) =;; l:fl dO'.
{f=c}
(27.75)
Both derivatives exist (note that IV f I > 0 on {f = e} since e is a regular value
of!).
PROOF. Since g and V^2 f = -Re +~g are both real analytic, we have that f
is real analytic. Therefore IV f 12 is real analytic. In particular, either IV fl^2 = 0 on
all of M or IV fl^2 = 0 only on a set of measure zero (see Krantz [171, p. 103]).
Since f is not a constant function, we have that IV fl-^1 is measurable. Hence, we
may apply the co-area formula (27.70) with h = IV fl-^1 and with h = RIV fl-^1
to the compact manifold with boundary {f :::; c} to conclude that, for any regular
value c E [O, oo) of f ~ 0, we have
(27.76) V(c) = ("de;; i;fidO',
lo {f=c}
(27.77) R (c) = re de;; ,:f I dO',
lo {f=c}
respectively. The lemma follows.
Integrating the formula (27.2) over {f < e} yields
(27.78) :!'.
2
V (e) - R (e) =;; 6.f dμ =;; ~f dO' =;; IV fl dO'
{f<c} {f=c} vi/ {f=c}
if e is a regular value of f. In particular,
(27.79)
n
R(c):::;
2
v(e).
By (27.6a), we have wherever IV fl -=!= 0 that
(27.80)
1Vfl
2
f-R
IV fl= IV fl = IV fl.
Applying this to (27.78) yields
LEMMA 27.32 (Differential identity relating V and R).
(27.81) ~ V ( c) - c V' ( c) = R ( c) - R' ( c) for a. e. e.
4.2. The AVR exists and is bounded.
D
For complete noncompact shrinkers, the AVR always exists. The result that
the volume growth is at most Euclidean is due to H.-D. Cao and D. Zhou, with
the assistance of Munteanu, who removed a technical condition that they assumed.
That the constant stated below depends only on n is due to Haslhofer and Muller.