302 35. STABILITY OF RICCI FLOW
because the hypotheses of the theorem require not merely curvature pinching, but
closeness in a c^2 +a (respectively, ci+.B) topology.
In the noncompact case, there h as b een considerable recent progress. Wu [438]
has studied the stability of complex hyperbolic space of two complex dimensions
and higher. To do so, h e constructs weighted little Holder spaces adapted to com-
plete noncompact manifolds. Using this technology, Williams and Wu [436] prove
dynamic stability under compact perturbations of t he homogeneo us solitons whose
linear stability was established by J ablonski , P etersen , and Williams (see Subsec-
tion 1.6 above).
3.3. Stability for positively curved metrics.
Motivated by the results above, it is natural to ask whether similar techniques
might provide an alternative proof of asymptotic stability for metrics of constant
positive sectional curvature. Consider normalized Ricci- DeTurck flow
8 2 -
(35.25) atg = -2Rc+;Rg + .Cwg
on a compact manifold Mn, where R = UM Rdμ)/UM dμ) and Wis the vector
field (35.4) that we encount ered in Subsection 1.2, with background metric g =
g (0). An Einstein metric g with Re = Kg for any co nstant K is a stationary
solution of (35.25). The linearization of (35.25) at the Einstein metric g is the
operator
(35.26) A: h H b.Lh + 2K { h - ~Hg}'
where H = tr 9 h and H =UM H dμ)/UM dμ).
Now let us make the stronger assumption that g is a metric of constant sectional
curvature k > 0, so that K = (n - l)k. By passing to a finite cover if necessary,
we may further assume that Mn is the round sphere s n embedded in IRn+^1. Then
one can make t he following observations, whose proofs may be found in [162]:
- A is linearly stable (but not strictly) if n = 2. Its null eigenspace is then
the (n + 2)-dimensional space
{cg : c E IR} U { xj g : 1 :::; j :::; n + 1},
where the (x^1 , ... , xn+l) are the standard coordinate functions of IRn+l. - A is linearly unstable for all n :'.'.'. 3. The sole unstable eigenvalue is
( n - 2)k with ( n + 1 )-dimensional eigenspace { xJ g : 1 :::; j :::; n + 1}. The
1-dimension al null eigenspace is {cg : c E IR}.
R ecall t hat the coordinate functions act as infinitesimal conformal diffeomor-
phisms (Mo bi us t ransformations). If h = f g for some f E { x1, ... , xn+l}, t hen
1h = -
2
k .Cgrad f (g).This shows that the linear instability of the operator A defined in (35.26) lies
only in the direction of diffeomorphisms; hence (like the linear instability of t he
cylinder soliton discussed above) it is "ungeometric". Round spheres are in fact
dynamically stable, as we discuss in Subsection 4.1 below. The unstable eigenspace
here is a consequence of the standard choice of DeTurck diffeo morphisms used to
impose parabolicity, rather than a feature of Ricci flow itself. Indeed, Ha milton has
observed [138] that t hose DeTurck diffeomorphisms solve a type of harmonic map