1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

16 2 7. NONCOMPACT G R ADIENT RICCI SOLIT O N S

By (27.42) and (27.46), we h ave good estimat es on the potent ial function:

(27.65) ~ [(d(x ,o)- c1)+r::; f(x )::; ~ (d(x,0)+2v'f(o))

2

for some const ant C 1. Choosing c > 0 sufficiently small , we h ave ¢ > 0 inside
B 0 (C1 + 3n). If infM-Bo (C,+3n) ¢ ~ -8 < 0, then by (27.65) there exist s p >

C 1 + 3n such that¢>-~ in M - B 0 (p). Thus a negative minimum of¢ is attained

at some point x 0 outside of B 0 (C 1 + 3n). By the maximum principle, evaluating

(27.64) at x 0 yields f(~o) - n ::; 0. However , (27.65) implies that f (x 0 ) ~^9 ~

2
, a
contradiction. We conclude that

R 2 cf -^1 + cnf -^2 on M.

The theorem now follows from (27.6 5 ). D

REMARK 27.27. M. Feldman , T. Ilmanen , and one of t he authors [111] con-
struct ed complet e noncompact K ahler shr inkers on the total sp aces of k-th powers

of tautological line bundles over the complex projective space c 1pm-l for 0 < k < n.

These examples, which have Euclidean volume growt h and quadratic scala r curva-
ture decay, show that T heorem 27. 26 is sh arp.

As a vari a tion on the proof of Theorem 27.26, define on {f > i } t he function
'ljJ (f) = 1 :_~, where c > 0. In general, we compute tha t

6. f (R - 'I/; (f)) = -2 JRc l
   2
   
   
   • R + (!-~) 'l/J' (f) - 'l/;
     11
     (f) IV fl
     2
     .
   
   
   
   

Since (f - i )'if/ (f) = -1/J (f) and 'I/;" 2 0, we obtain

(27.66) 6.1 (R - 'I/;(!))::; -2 JR c j^2 + R - 'I/; (f).

Choose c sufficient ly sm all so that R 2 c on {f = i + 1 } , which implies tha t

R - 'ljJ (f) 2 0 on {f = i + l}. By applying the maximum principle to (27.66),

since liminfx-+oo(R - 'ljJ (f))(x ) 2 0, we obtain a contradiction if R -'ljJ (f) < 0

somewhere in {f 2 i + l}. Hence R - 'ljJ (f) 2 0 in {f 2 i + l}.

3.2. Lower estimate of R for steadies.
By a similar argument to the previous subsection we may prove the following
res ult, due to B. Yang and two of t he aut hors, rega rding steady GRS assuming a
condition on the potential function.

THEOREM 27.2 8 (Scala r curvature of nonflat steadies decay at most exponen-

tially ). L et (Mn , g , f ,O) be a complete normalized st ea dy GRS. Iflimx-+oof (x ) =

• oo and f ::; 0, then R 2 A+ 2 ef. Since IV f I ::; 1, this impli es that for any

6 EM we have

(27.67) R(x )^2 ce-d(x,O) for all x EM,

where c = (\/1' + 2)-^1 ef(O)_

PROOF. T he idea is to find a lower barrier function (an expression of f) for

the scalar curvature R. Using (27.9b ), we compute that

b.1 (e^1 ) = e^1 b.tf + ef 1Vfl^2 = -Re^1 < 0.