306 35. STABILITY OF RICCI FLOWThe same authors also prove an instability result that demonstrates that the
local maximizer hypothesis of the above theorem is natural. (As noted above,
Sesum's integrability condition is a special case of this.)THEOREM 35.42. L et (Mn, g) be a compact Ricci-fiat manifold. If (g) is not a
local maximizer of>., then there exists a nontrivial ancient Ricci flow {g(t) }tE(-=,OJ
that converges (modulo diffeomorphisms) to g fort-+ -oo.
4.3. Dynamic stability of noncompact flat space.
Lang-Fang Wu has shown that under certain conditions on the initial data,
Ricci flow originating at a complete metric on JR^2 exists for all time and approaches
a soliton solution [440]. An alternate approach has been used by Schniirer, Schulze,
and Simon, who consider C 0 -small perturbations of Euclidean space [346]. They
show that if the space is asymptotically Euclidean, then solutions of a coupled Ricci-
harmonic heat flow exist for all time and converge uniformly to the fiat Euclidean
metric. Some of their results have been improved by Koch and Lamm [165], who
prove the existence of global analytic solutions of Ricci- DeTurck flow on Euclidean
space for bounded initial metrics that are L = close to the fiat Euclidean metric.
4.4. Dynamic stability of negatively curved spaces.
Ye has studied asymptotic stability of Ricci flow [449] for certain geometries.
He considers compact manifolds (Mn,g) with n 2 3 on which the self-adjoint
elliptic operator(
L: h iJ H D..LhiJ + Rj k + -;;,Rgj^1 k) hik
has strictly negative spectrum. This occurs, for example, at a metric of negative
sectional curvature whose Ricci eigenvalues are sufficiently pinched, or at a metric of
negative scalar curvature whose sectional curvatures are sufficiently pinched. Ye's
approach requires certain hypotheses on diameter, volume, and the L^2 -norm of the
(traceless) Ricci tensor- but does not require a priori knowledge of the existence
of a hyperbolic metric.
In dimensions n 2 6, Li and Yin prove stability of the constant curvature metric
on IH!n under small perturbations that decay sufficiently quickly at spatial infinity
[186]. (Smallness here is measured by a technical condition related to Ye's work.)
Schniirer, Schulze, and Simon prove a similar result for coupled Ricci-harmonic map
flow, assuming perturbations that are bounded in L^2 and small in c^0 [347]. More
generally, Bamler establishes stability for noncompact symmetric spaces evolving by
Ricci flow [18]; this strengthens the Schniirer, Schulze, and Simon result for IH!n by
imposing weaker assumptions on the perturbation. Bamler also proves that finite-
volume noncompact hyperbolic manifolds of dimensions n 2 3 are stable under a
rescaled Ricci flow, without imposing any decay on the perturbation [17].