1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

4. 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.