l. BASIC PROPERTIES OF GRADIENT RICCI SOLITONS 7THEOREM 27.4 (Bounds for IV'fl). Suppose that (Mn,g, f,c) is a complete
normalized GRS. Then, given any 6 EM, we have the following:
(1) For a steady,(27.29) l\i'fl (x)::; 1 for all x EM.
(2) For a shrinker we have f 2 0 and for all x EM,
(27.30) l\7 fl (x)::; f^1 l^2 (x)::; f^112 (0) + ~d(x, 0).
(3) For an expander we have f::; ~ and for all x EM,
(
n )1/2 1 _ (n _ )1;2
(27.31) IV'fl(x):S 2-f(x) :S2d(x,0)+ 2-f(O).PROOF. (1) c = 0. Since R 2 0, by (27.6b) we have
(27.32) 1 = R + l\7f1^2 2 l\7f1^2.
(2) c = -1. Again since R 2 0, by (27.6a) we have
(27.33) l\7 fl^2 = -R + f :Sf.
For any x E M, let 'Y : [O, d(x, O)] -+ M be a unit speed minimal geodesic joining
6 to x. The function F ( s) ~ f ('Y ( s)) satisfies
dF
ds (s) = \7 f b (s)) · 'Y
1
(s) :S l\7 fl ('Y (s))::; F^1 /^2 (s).Integrating this over [O, d(x, O)], we obtain^2F^112 (d(x, 0))::; F^1 /^2 (0) + ~d(x, 0).
Because "f(d(x, 0)) = x, this and (27.33) yield (27.30).
(3) c = l. Since R 2 -~,we have
(27.34) -f = R+ l\7fl^2 2 -~ + l\7fl^2 -
Let the geodesic 'Y be as in part (2). The function G (s) ~ -f ('Y (s)) + ~ 2 0
satisfies
dGds (s) = -\7 f ('Y (s)). 'Y^1 (s) ::; l\7 fl ('Y (s)) ::; c^1 /^2 (s)
by (27.34). Therefore G^112 (d(x, 0)) ::; G^112 (0) + ~d(x, 0), which implies (27.31).
D
REMARK 27.5. The example of the Gaussian soliton, where we have l\7 fl (x) =
J¥ lxl for x E ~n, shows that the above upper bounds for l\7 fl are qualitatively
sharp.
The following is elementary.
LEMMA 27 .6 (Criterion for completeness of a vector field). Suppose F : [O, oo) -+
[O, oo) is a locally Lipschitz function with the property that any solution u ( t) to theODE!~ =F(u), withu(O) E [O,oo), existsforalltE [O,oo). IfavectorfieldX on
a pointed complete Riemannian manifold (Mn, g, 0) satisfies IXI (x) ::; F(d(x, 0))
for all x E M, then X is complete.
(^2) Foreach8>0wehave fs((F(s)+8) (^112) )::; !·