1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

4. VOLUME GROWTH OF SHRINKING GRADIENT RICCI SOLITONS 17

By this and (27.10), we obtain for c E JR,

2 nc^2

6. 1 (R - eel) ::::; --;;,R^2 + cRel::::;
   8

e^2 I.

Using 6. 1 (e^2 1) = 2e^21 (1 - 2R), we compute for any constant b E JR tha t

(27.68) (nc

2

6.1 (R - eel - be^2 1) ::::; S -2b + 4bR ) e^2 1.

Given b, c > 0 to be chosen below, suppose that R-cel -be^21 is negative somewhere.

Then, since R 2 0 by Theorem 27.2(1) and since limx->oo el(x) = 0 by hypothesis,

a negative minimum of R - eel - be^21 is attained at some point. By (27.68) and

the maximum principle, at such a point we have

nc^2 nc^2

0 :S S -2b + 4bR < S -2b + 4b ( c + b)

since f::::; 0. Given c E (0 , ~], the minimizing choice b =^1 ~^2 c yields (i-,;cl

2

< nf.

With this choice of b, we obtain a contradiction by then choosing c = vfi+ 2. 0

R 2729 D h · l' (m>2 dx

(^2) +d1/) h R I h
EMARK.. ror t e cigar so 1ton m. , i+x 2 +y 2 we ave = e , w ere
f ( x, y) = - ln ( 1 + x^2 + y^2 ).

4. Volume growth of shrinking gradient Ricci solitons

In this section we discuss the asymptotic volume ratio of noncompact shrinkers,

including a Euclidean upper bound for their volume growth. We consider two

approaches: sublevel sets of the potential function and the Riccati equation along

geodesics.

4.1. A differential identity for the volume ratio of sublevel sets.

Given a complete noncompact Riemannian manifold (Nn, h) and a basepoint

6 EN, the asymptotic volume ratio (AVR) is defined by

(27.69) AVR(h) ~ lim VolBa(r),

r->oo Wnrn

provided the limit exists, where wn is the volume of the unit Euclidean n-ball. If
Re 2 0, then AVR(h) E [O, l] exists by the Bishop volume comparison theorem. In
general, whenever the AVR exists, it is independent of 6.
In the sublevel set approach, we shall need the co-area formula (see Schoen and
Yau [355, p. 89] or Lemma 5.4 of [77] for example), which says the following.

PROPOSITION 27.30 (Co-area formula). Let (Mn, g) be a compact manifold

with or without boundary. If f is a Lipschitz function and if h is an L^1 function

or a nonnegative measurable function, then

j hl'Vfldμ=1

00

def hdu,

M -oo {/=c}

(27.70)

where dμ is the Riemannian measure on M and where du is the induced measure

on {f = c}.