1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

18 27. NONCOMPACT GRADIENT RICCI SOLITONS

Henceforth, let (Mn, g, f, -1) be a complete normalized noncompact shrinking

GRS structure. Extend this structure to a complete solution g (t), t E (-oo, 1),

to the Ricci flow with g (0) = g. By Remark 13.32 in Part II and by the local

nature of real analyticity, we may extend Bando's Theorem 13. 21 in Part II to the

noncompact case. Namely, using Shi's local derivative estimates, one can show that

M has a real analytic structure and that, given t , the metric components 9ij ( t) are

real analytic functions in any normal coordinate system. In particular, g is real

analytic. We shall use this fact in the sublevel set approach.

By (27.46) we have that f attains its minimum at some point 0 E M. Thus,

from (27.2), we have

(27.71)

n n

inf R(x)::::; R(O) = - - b.f(O)::::; -.

xEM 2 2

We claim that, since M is noncompact, we have infxEM R(x) < ~· If not, then on

M we have b.f = ~ - R ::::; 0. Since f is superharmonic and attains its minimum,

we conclude that f is a constant function by the strong maximum principle, which

contradicts M being a noncompact shrinker.

Define the functions

by3

(27.72a)

(27.72b)

v : JR --+ (0, 00) )

R : JR --+ (0 , oo)

V(c)~J dμ=Vol{f<c},

{f<c}

R(c) ~ J Rdμ

{f<c}

for c E JR, which are nondecreasing nonnegative functions of c since R ~ 0.

Since we have good control of f , to approach the A VR we shall use ~J~1 instead

of the volume ratios of geodesic balls. More precisely, by (27.42) and (27.46), given

any 6 E M, there exists a constant C 1 < oo such that

1 ( 4 (d(x, 0) - - C1)+ )2 ::::; f (x)::::; 1 ( - )2

4

d(x, 0) + C 1

Therefore, if r ~ C 1 , then

{ f < ~(r -C1)

2

} c B 0 (r) c { f < ~ (r + C1)

2

}.

Thus

V(i(r - C1)^2 ) VolB 0 (r) V(i (r + C1)^2 )

~--'----~~~-< <.

rn - rn - rn

Thus limc--roo ~f ~l exists if and only if AVR(g) exists. Thus, in this case,

(27.73) 2nwn AVR(g) = lim V(/c

2

).

c--roo en

Recall that by Sard's theorem, the set of singular values of f has Lebesgue

measure zero. Note that in general 8 {! < c} C {f = c }, whereas for a regular

value c of f we have that 8 {! < c} = {f = c} is a smooth compact hypersurface

(^3) Note the distinction between the function R and the scalar curvature R , whose notations
look similar.