1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

3. DYNAMIC STABILITY RESULTS OBTAINED USING LINEARIZATION 303

fl.ow with respect to a domain metric evolving by Ricci fl.ow and a fixed target metric

(in this case, the stationary solution of Ricci fl.ow about which one is linearizing).

It is well known [100] that the identity map of round spheres id : sn ~ sn is a

weakly stable harmonic map for n = 2 and an unstable harmonic map for all n 2". 3.

3.4. More dynamic stability results obtained by linearization.

In [163], Young and one of the authors apply maximal regularity theory to the

fully nonlinear cross-curvature fl.ow, which has been developed as an alternate to the

Ricci fl.ow in the hope that it would give more information about negatively curved

3-manifolds. Given a manifold (M^3 , g) with strictly negative sectional curvature,

the fl.ow is defined to be

(35.27) ag = -2X

at '

where Xij = ~pμv Riuvj and Pab =Rab-~R9ab is the Einstein tensor. The authors

prove that hyperbolic metrics are stable stationary solutions of a normalized cross-

curvature fl.ow (as usual, obtained by rescaling in space and time to obtain a fixed

point). They also prove stability of constant curvature hyperbolic manifolds under

Ricci fl.ow. The analysis uses maximal regularity theory and an asymptotic stability

theorem for fully nonlinear systems [92].

In [130], another of the authors together with Todd Oliynyk applies the ideas

of maximal regularity to the fully nonlinear second-order renormalization group

fl.ow

8 a 2

atg = -2Rc-

2

Rm.

(Here, a is a fixed positive constant.) In the fl.at case, the techniques from Ricci

fl.ow carry over in a straightforward way. For the hyperbolic case, one wishes to

rescale the equation such that the metric is a fixed point. In this equation, however,

the nonlinear term scales like Rm^2 (cg) = ~ Rm^2 (g); consequently after rescaling,

the equation is no longer autonomous. Nevertheless, working instead with a related
system, one does obtain a stability result for the rescaled equation, for a sufficiently
small values of the parameter a.
In [364], Sesum proves that for Ricci fl.at metrics, linear stability together with
an integrability condition implies dynamical stability. The integrability condition
essentially says that infinitesimal Ricci-fl.at deformations have a smooth manifold
structure near the metric about which one linearizes. As described below in Sub-
section 4.2, Haslhofer and Muller later showed that the integrability condition is a
special case of being a maximizer of Perelman's .A functional [147]. Sesum's results
have been used by Dai, Wang, and Wei to prove that Kahler- Einstein metrics of
nonpositive scalar curvature are dynamically stable [89].
If an immortal solution of Ricci fl.ow fails to converge, the failure typically man-
ifests itself in collapse, with the injectivity radius going to zero while the curvature
remains bounded. Using the machinery of etale groupoids, Lott [209] has proven
that in many cases, properly rescaled solutions converge to solutions on bundles

Q y M ~ B, with Q a nilpotent Lie group and Bn compact and orientable. The

structure of these bundles reflects the dimension-reduction of the original solution,
with the fibers Q yielding geometric information about its collapsed directions. One
of the authors has proven that if Q is abelian (which is always true for 3-dimensional