l. LOCAL ESTIMATE FOR THE SCALAR CURVATURE UNDER R ICCI FLOW 39COROLLARY 28.2 (Ancient solut ions h ave nonnegative scalar curvature). If
(Mn,g(t)), t E (-oo,O], is a complete ancient solution to th e Ricci flow, th en
(28.21) R :'.'.'. 0 on M x (-oo, OJ.
As a further corollary, we obtain Theorem 27 .2 since we may use Theorem 4.1
in [77] to put a co mplete GRS in canonical form. For example, both shrinking
and st eady GRS a re a ncient solutions and hence h ave R :'.'.'. 0. Likewise, for anexpander metric g with E > 0, the corresponding solution g (t) with g (0) = g
exists on the time interval ( -~, oo) , so that R (g ( t)) :'.'.'. - 2(t: ~) and in particular
R(g) = R(g(O)) :'.'.'. -~"·
We may attempt to use the same localization method as in the proof of Theorem- l to study the following.
PROBLEM 28 .3. Let (M^2 , g (t)), t E (0, T), be a complete so lution to the Ricci
fl.ow with positive scalar curvature on a noncompact surface. Does the estimate
. f) 2 1
Q =:= fJt lnR - l\7 lnRI :'.'.'. -t
hold?
The issue is that we do not assume that the scalar curvature is bounded. Note
tha t
~~ :'.'.'. 6.Q + 2 \7 ln R · 'VQ + Q^2.
In attempting to localize t his formula , we note that the gradient term 2\7 ln R · 'VQ
poses a problem.
Rega rding incomplete an cient so lutions, Peter Topping h as pointed out to us the
following examples of incomplete a ncient so lutions with scalar curvature negative
somewhere.
EXAMPLE 28.4. Consider the 2-dimensional hyp erbolic disk ( D^2 , 91HI) and mod-
ify the metric 9 1HI only in the ball of radius 1 centered at the origin 0 (measured
with respect to 9 1HI) so that the resulting metric go on D^2
(1) is rotationally symmetric,
(2) h as nonpositive curvature everywhere,
(3) is fiat in the b all of radius 1/ 10 (measured with resp ect to the fiat metric)
centered at 0.Then , by W.-X. Shi's short-time existen ce theorem, there exists a complete solut ion
( D^2 , g ( t)) to the Ricci fl.ow satisfying the following properties:
(i) g(t) is defined fort E [O,c) for some c > 0,
(ii) g (0) =go,
(iii) g (t) h as bounded curvature for all t E [O, c).Since ~~ = 6.R + R^2 (we are here in dimension 2), it can be proved by the weak
and strong maximum principles t hat -2 < R (g (t)) < 0 for all t E (0 , c). If we let
B ~ B 9 (o) (0, 1/ 10 ), then (B, g (t)l 8 ) , t E [O,c), is an incomplete solution to the
Ricci fl.ow with -2 < R (g (t)l 8 ) < 0 fort E (0, c) and R (g (0) 18 ) = 0. Defining
the metric on B at all t imes t < 0 to be identically equal to g (0)1 8 , we obtain an
inco mplete ancient solution with scalar curvature equ al to 0 for all t ::::; 0 and scala r