1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

2. ESTIMATES FOR POTENTIAL FUNCTIONS OF GRADIENT SOLITONS 13

Since the assumption (27.54) is on the edge of holding true in the sense that it

is true for a = ~ (see Lemma 27 .15), we ask the following.

PROBLEM 27 .17 (Conjecture 1.3 in [105]). Can one remove the condition

(27.54) with a < ~ in Corollary 27 .16?

A solution to Problem 27.17 would follow from obtaining a good lower estimate

for IV' 112 , i.e., one which would show that IV' 112 is positive outside a compact set.

More ambitiously, one may pose the following:

OPTIMISTIC CONJECTURE 27.18. For any complete noncompact shrinking GRS,

the scalar /Ricci/sectional curvature is necessarily bounded from above.

One of the best results in this direction is due to 0. Munteanu and J. Wang;

they proved the following.

THEOREM 27.19. Let (M^4 , g , f , -1) be a 4 -dimensional complete noncompact

shrinking GRS. If the scalar curvature R is bounded, then there exists a constant

C < oo such that IRml :::; CR on M. I n particular, IRml is bounded.

Some elementary evidence for Optimistic Conjecture 27.18 is given by inequality
(27.79) below, which says that the average scalar curvatures on the sublevel sets of
the potential function are bounded above by ~.
Next we improve the scalar curvature upper bound for a net of points. We say
that a countable collection of points {xi} in a Riemannian manifold (Mn, g) is a
J-net if for every y E M there exists i such that d (y, Xi) :::; o. The following result
supports Problem 27.17.

LEMMA 27.20. Let (Mn, g , f , -1) be a complete normalized noncompact shrink-

ing G RS and let 0 E M be a minimum point of 1. Then for any o > 0 there exists

a constant C ( n , o) < oo such that for ·any x E M - Bo ( C ( n , o)), there exists

y E Bx ( o) such that

(27.56) R(y):::; C(n,o) (d(y, O) + 1).

In other words, the scalar curvature has at most linear growth on an o-net, where

the rate of growth depends on o.

PROOF. Define

s

8

1

-8-r(x)- s

ifs E [O, J],

if s E ( o, r ( x) - o],

ifs E (r(x)-o,r(x)].

As in (27.50), we obtain from (27.49) that

2(n-l) r(x) 2J 21

8

--"-----'-> --- - + -s (f 0 "Y )'d s - 2 1r(x) r(x)-s(f 0 "Y )'d s.

0 - 2 3 0 J2 r(x)-8 J2

Using I(! o "Y)' (s)I :::; VJ (0) + ~ for s E [O, J], the above formula implies that

2Jr(x) r(x)2-s(fo"Y)'ds2'. r(x) _2(n-l+~)-Jf(O).

r(x)- 8 0 2 0 2