3. DYNAMIC STABILITY RESULTS OBTAINED USING LINEARIZATION 301

and hence estimate (35.23) by

(Lh, h) = --^1 2 2 r^2 n-2^2 J

2

llTll - llbhll - - litr_ghll - -r llhll + W(h, h) dμ

n-l n - l M"

(35.24) :::; --n-2 r llhll^2 + j W(h, h) dμ.

n - l M"

Now suppose that n > 2 and that (Mn, g) has constant negative sectional cur-

vature. Then the Weyl curvature W of g vanishes, and g is a stationary so lution

of the dilated Ricci fl.ow (35.21). Calculation (35.24) shows that g is strictly lin-

early stable. (The null eigenvalue that appears for n = 2 reflects the presence of

Teichmiiller space as a center manifold.) For n 2: 3, Mostow rigidity ensures that

g is unique up to scaling, hence that the choice of normalization in (35.21) forces a

unique fixed point.

Once we have (35.24) and the consequent linear stability of g, it follows for n > 2

that the argument presented in Proposition 35.19 implies asymptotic stability of the

modified Ricci fl.ow (35.21), modulo DeTurck diffeomorphisms, for all initial data

in a 2+a little Holder neighborhood of a hyperbolic metric g, where 0 <a< 1 (see

Theorem 9.1.2 or Theorem 9.1.5 in [214]).^6 Because all nearby solutions converge

exponentially fast to the unique fixed point of dilated Ricci fl.ow (there is no center

manifold present), it is easy to verify that the DeTurck diffeomorphisms converge

exponentially fast as well ; hence, every solution g( T) of the modified Ricci fl.ow itself

converges exponentially fast in T (see Lemma 35.28 and Corollary 35 .29). Moreover,

the relation

g(t) = O'(t)g(T(t))

shows that the solution g(t) of unnormalized Ricci fl.ow obtained from g(T) exists

for all time and its sectional curvatures become spatially homogeneous. Thus we

have proved the following:

THEOREM 35.31. Let (Mn, g) be a compact Riemannian manifold with n > 2

and constant sectional curvature K < 0. Then there exists 6 > 0 such that for

all initial data g 0 in a little Holder ball B~+a (g) around g, the unique solution

g(t) of normalized Ricci flow satisfying g(O) =go exists for all time and converges

exponentially fast to a constant curvature metric (which takes the form >..g for some

>.. > 0 since all constant negative curvature metrics are proportional to each other).

REMARK 35.32. By refining the techniques of the theorem, it is possible to
show that for any initial data in a 1 + (3 little Holder neighborhood of g, Ricci fl.ow
converges tog in the 2 +a little Holder topology, where (3 E (a, 1) depends on a.
This result takes optimal advantage of the parabolic smoothing properties of t he
quasilinear nature of Ricci fl.ow. See [162] (by one of the authors) for details.

REMARK 35.33. Farrell and Ontaneda [107] prove that for every n > 10 and€>

0, there exists (Mn, g) such that Mn is compact, g is hyperbolic, and there exists
a Riemannian metric hon Mn whose sectional curvatures lie in [-1 - c:, -1 + c:]
but such that the Ricci fl.ow starting at h cannot converge smoothly to g. Their
result is compatible with Theorem 35.31 (and its refinement in the remark above)

(^6) Maximal regularity theory is not necessary for this result, but it simplifies the proof and
in general allows weaker regularity assumptions on the (nonlinear) right-hand side. We refer the
reader to Chapters 8 and 9 of [214] for details.