l. LINEAR STABILITY OF RICCI FLOW 285where ei is an orthonormal coframe. Then the Lichnerowicz Laplacian satisfies the
formula
'.D*'.D(<l>(h)) = <l>(-b.Lh),
where '.D is the Dirac operator.
In light of this result it is interesting to ask which compact Ricci flat manifolds
admit nonzero parallel spinors. As a consequence of the Cheeger- Gromoll splitting
theorem, it suffices to consider simply-connect ed , comp act, Ricci-flat manifolds.
One may also assume the manifold is irreducible.^3 Then one h as the following:PROPOSITION 35.10 ([88]). A simply-connected, irreducible, compact, and Ricci-
fiat manifold (Mn,g) either has holonomy SO(n) or admits a nonzero parallel
spinor.
REMARK 35.11. All known examples of compact Ricci-flat manifolds have sp e-
cial holonomy and so admit nonzero parallel spinors.
Consequently, there are no known compact Ricc i-flat manifolds that are linearly
unstable.
In anot her paper, D ai, Wang, and Wei prove a related result:
THEOREM 35. 12 ([89]). If a compact Einst ein manifold (Mn,g) with nonposi-
tive scalar curvature admits a nonzero parallel spine spinor, then the linear operator
A: hij H b.Lhij + R~hkj + Rjhik
has nonpositive spectrum. In particular, A has nonpositive spectrum on any compact
Kahler-Einstein manifold with nonpositive scalar curvature.
Note that a simply-connected manifold admits a nonzero parallel spine spinor
if and only if it is the product of a Kahler manifold and a manifold that admits a
nonzero parallel spinor.1.6. Nontrivial Ricci solitons.
As seen above in Subsection 1.2, every nontrivial (i.e. non-Einstein) Ricc i soliton
is a stationary solution of a dilated Ricci flow
8
otg = -2Rc+2.Ag + .Cxg.It is natural to consider t he linear stability of such stationary solutions.
The cylinder soliton is linearly unstable, but the unstable perturbations are
merely "coordinate instabilities" rather than geometrically meaningful ones. For
example, it follows from the maximum principle that a slightly smaller cylinder
vanishes under Ricci flow quicker than a slightly larger one does. Nonetheless, both
remain round cylinders under Ricc i flow. (See [14] for more details.) We discuss
this phenomenon further in Subsection 4.1 below.
A more interesting class of examples is the noncompact exp anding homogeneo us
solitons, which in some cases model immortal solutions of Ricc i flow that collapse
with bounded curvature (see, e.g., [209]). An algebraic soliton is defined tob e a left-invariant metric g on a homogeneous space gn that satisfies the equation
Re = .AI +D, where Re is the Ricc i endomorphism of g, .A E IR, and Dis a derivation
(^3) Here "irreducible" means "not locally a product space" in the sense of the de Rham decom-
position of T* Mn.