1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

4 27. NONCOMPACT GRADIENT RICCI SOLITONS

for some universal const < oo. We define <Pc : M ~JR by

(27.12) <Pc (x)=rJ (r(- c-x)) R(x).

Throughout the proof, c E [2, oo); eventually we let c ~ oo.
Taking the f-Laplacian of (27.12), we have
2 (ry'o:C: r/'o:C: )

ll1<Pc = ( 'r) o ~) ll1R + ~ ( 'r/

1

o ~) ('vr, V' R ) + ~ll1r + ~ IY'rl

2

R.

Applying (27.10) and IY'rl^2 = 1 to this, while dropping " o ~" in our notation, we
have
(27.13)

ll1<Pc = ry(-2 IRc l^2 - c:R) + -2r/ (V'r, V' R) + ( -ll1r rJ' + 2 ry") R

c c c

( I 12 ) 2ry' ( ) r/ ( )^1 ( /1 ( 'r/

1

)

2

= 'r) -2 Re - c:R + - \i'r, Y'<Pc + - ll1r R + 2 'r) - 2--) R

C'r) C C 'r)

at all points where 'r) =/= 0.

STEP 2. Applying the maximum principle^1 to <Pc. Now suppose that there
exists Xe E M such that

(27.14) <Pc (xc) =min <Pc < 0.

M

Otherwise, we have R 2 0 in all of Ba (c).

Applying t he first and second derivative tests to (27.13), using 1Rcl

2

2 ~R^2 ,

and dividing by R (xc) < 0, we h ave that at Xe,

(27.15) 0 2 'r/ (-3-R-c:) + ry' L:l 1 r + 2_ ('r/" - 2 (ry')

2

).

n c c^2 'r/

We consider two cases, depending on the location of Xe.

Case (i): Xe E Ba (c). Then ryo ~ = 1 in a neighborhood of Xe, so that (27.15)

and (27.14) imply

(27.16) 0 2 --R(x^2 c) - € = --<I>^2 c (x c) -€ 2 - -ry^2 (r(- x)) R(x) -c:

n n n c

for all x E M. This yields the estimate

nc:

(27.17) R (x ) 2 -2

for all x E Ba (c) since 'rJ o ~ = 1 in Ba (c).

Case (ii): Xe ¢:. Ba (c). Regarding (27.15), since ry' ::; 0, we wish to estimate
the t erm ll1r from above. Recall from Lemma 18.6 in Part III (or the original
Lemma 8.3 in P erelman [312]) that

(27.18)

r(xc)

llr(xc)::; Jo ( (n - 1) ((')

2

(s ) - (^2 Re ('y'(s), "f^1 (s))) ds

(^1) The distance function r (x) is in general only Lipschitz continuous. When applying the
maximum principle, to address the possible nonsmoothness of r (x), one may use Calabi's trick;
see p. 395 of [77] or pp. 453-456 in Part I for example.