- LOWER BOUNDS FOR R OF NONFLAT NONEXPANDERS 15
We have the following results of Y. Deng and X. Zhu.THEOREM 27.24. Let Q = (M, g, f , 1) be an expanding Kahler GRS with dime M
= n and let 6 EM. If RE o(d-^2 (x,O)), then (M,g) is isometric to en.
COROLLARY 27.25. There does not exist an expanding Kahler GRS with dime M?: 2 and Re?: ryRg, where 'T/ > 0 and R > 0.
3. Lower bounds for the scalar curvature
of nonfiat nonexpanding gradient Ricci solitons
In this section, in the case of nonflat nonexpanding (i.e., shrinking and steady)
GRS and with the aid of the estimates for the potential function of the previous
section, we sharpen the lower bounds for the scalar curvature given in §1.3.1. Lower estimate of R for shrinkers.
In this subsection we discuss a lower bound for the scalar curvatures of non-
compact nonflat shrinkers. We have the following result due to the combined works
of B. Wilking, B. Yang, and three of the authors.THEOREM 27.26 (Scalar curvature of nonflat shrinkers decay at most quadrat-ically). Let (Mn, g, f, -1) be a complete normalized noncompact nonfiat shrinking
GRS. Then for any point 6 EM there exists a constant Co > 0 such that
R(x) ~ C 01 d-^2 (x, 0)wherever d(x, 0) ~ Co. Consequently, the asymptotic scalar curvature ratioASCR (g) > 0 (see (19.8) in Part III for its definition) and the asymptotic cone, if
it exists, is not fiat.PROOF. We modify the quantity R appearing in (27.10) by adding powers ofj. For any p E JR, using (27.9a) we compute that
(27.60) fj.1 (~rp) = rp -f-p-l ( ~ -(p+ 1) ivp
2
).In particular, we shall use the formulas (p = 1, 2)
(27.61) fj.f u-1) = f-1 -r2 ( ~ - 21vp2),
(27.62) fj.f u-2) = 2f-2 - f-3 ( n - 61vp2).
Using (27.10) and (27.61), we compute for any c > 0 that
(27.63) f).f (R - cf-^1 ) ~ R - cf-^1 + cf-^2 ( ~ - 21\7 J1
2
).
Keeping in mind that we wish to modify (27.63) so that more negative terms appear
on the RHS, we define </> =i= R - cf-^1 - cnf-^2. By (27.62), we obtain
(27.64) lj.J<f> ~ </>-cnr^3 (f-n )-cr^4 (2f + 6n) 1Vfl
2
.