- DYNAMIC STABILITY RESULTS OBTAINED USING LINEARIZATION 299
Thus far in this argument, we have shown that there exists a neighborhoodB ( X, g, 5) such that the Ricci- DeTurck fl.ow g ( t) starting at any g 0 E B ( X, g, 8)
becomes fiat exponentially fast in the [[· [l 2 +P-norm for as long as g (t) E B (X, g, 8).
Since [Rij[, [\7j9ke - Oj9ke[, and [\7i\7j9ke - OiOjlJkel all decay exponentially fast ifg (t) EB (X, g, 8), there are C = C (8) < oo and w = w (8) > 0 such that
I :tgl = [-2 Ric-P_g(g)[::::: ce-wt
for as long as g (t) remains in B (X, g, 8). If we now choose 0 < E < 8 small
enough so that C (c) /w (8) < 8 - E, it then follows that for all solutions g (t) with
g 0 EB (X, g, c), we can estimate
[g (t) - g[ ::; [g (t) -go[+ [go - g[ < 8 - E + E = 8
independently of t ;::: 0. Bounds on the derivatives of g are shown similarly. It
follows that g ( t) remains in B ( X , g , 8) for all time and hence converges to a unique
fiat metric. By Corollary 35.29, the same is true of the unique solution of the Ricci
fl.ow with the same initial data. D
3.2. Dynamic stability in the presence of negative curvature.
The situation regarding negatively curved metrics is considerably more delicatethan the positively curved case. For example, for every n ;::: 4 and E > 0, Gromov
and Thurston [126] prove that there exists a closed n-manifold that admits a metric
with pinched sectional curvatures -1 - E < K ::; -1 but no metric with K = -1.
Moreover, Farrell and Ontaneda [108] show that if n;::: 10 and Mn admits a metric
with negative sectional curvatures, then the space of negatively curved metrics on
Mn has infinitely many path components. (Also see [107].) So it may be impossible
to deform a given negatively curved metric continuously to an Einstein metric of
negative sectional curvature.
However, it is possible to establish both linear and asymptotic stability for
a modified fl.ow at Einstein manifolds with negative scalar curvature, if one as-
sumes closeness to the hyperbolic metric in an appropriate norm, and not merely
a curvature-pinching condition. (See Subsection 4.4 below for related work of Ye
obtained using a very different method.) We now discuss this approach more pre-
cisely.
Suppose (Mn, g(t)) is a solution of unnormalized Ricci fl.ow that exists for
- oo <To < t < T 1 ::; oo. Given any constant r > 0, we define a scaling factor
O'(t) = 2r(t - To)
and a rescaled time variable
1
T(t) =
2
r log(t - To),noting that dT/dt = l/O'(t). Defining a 1-parameter family of metrics g(T) on Mn
via
g(t) = O'(t)g(T(t)),
i.e., g(T) = O'(To + e^2 rr)-^1 g(T 0 + e^2 r^7 ) , we see that the family g(T) is equivalent
to g(t) modulo rescaling of space and time. Moreover, g(T) evolves by the dilated
Ricci flow
(35.21) :
7
g = -2Rc-2rg.