14 27. NONCOMPACT GRADIENT RICCI SOLITONS
Since 2 J;(~/-li r(~~-s ds = 1, there exists s E [r (x) - cl, r (x)] such that y ~ 1(8)
satisfies
IV' fl (y) > r(y) - 2 (n -
1
- ~) - {i
- 2 cl 2 V2
since r(x) 2: r(y) and VJ (0)::::; A· Thus
2 1 2
IV' fl (y) ;::::
4
((d (y, O) - 2C(n, cl))+) ,where C(n, cl)~ 2(n8l +%)+A·
On the other hand, by (27.44) we have
1
f(y)::::: 4(d(y,0)+~)^2.Therefore, if d (x, 0) ;:::: 2C(n, cl)+ cl, then d (y, 0) 2: 2C(n, cl) and hence
R(y) = f(y) - IV' fl^2 (y)
1 1:S 4(d (y, 0) + ~)^2 -
4
(d (y, 0) - 2C(n, cl))^2
= d(y, O) ( /% + C(n, cl)) + ~ -C(n,cl)
2
.This motivates us to considerD
CONJECTURE 27.21 (Elliptic Harnack estimate for the scalar curvature). Let(Mn, g, f , -1) be a complete noncompact shrinking GRS. There exists canst < oo
such that for any x, y EM with d (x, y) ::::; 1 we have
(27.57) R(x)::::; constR(y).EXERCISE 27.22 (An elliptic Harnack estimate would imply finite topological
type). Show that the truth of (27.57) would affirm Problem 27.17.
Returning to the lower bound for IV' fl of a noncompact shrinker, note that the
most optimistic conjecture would be that IV' fl (x) 2: ~d(x, 0) -C on M for some
constant C. By (27.6a) and (27.46), we have
(27.58)
so that such an estimate would follow from a uniform upper bound for R.
Regarding the scalar curvature of a noncompact steady or expanding GRS, if
we further assume a positive Ricci pinching condition, then R attains its maximum
and has quadratic exponential decay (see D. Chen and L. Ma [215] or Theorem
9.56 in [77]). In particular, we have
PROPOSITION 27.23. Let Q = (Mn, g , f, 1) be an expanding GRS with Re 2:
r]Rg, where rJ > 0 and R > 0. Then, for any fj < rJ, there exists C < oo such that
(27.59)