284 35. STABILITY OF RICCI FLOW
1.3. Flat geometries.PROPOSITION 35.5. If (Mn, g) is fiat, then 6.L is negative semidefinite on 52
with its kernel consisting exactly of parallel (2, 0)-tensors; hence, the dimension of
the kernel is at most n ( n + 1) /2. In particular, any fiat metric is linearly stable
for the Ricci- DeTurck flow.
PROOF. This follows from the fact that 6.L = 6. on a flat manifold and from
the nonpositivity of the eigenvalues of 6.. D
1.4. K3 complex surfaces.DEFINITION 35.6. A K3 surface is a closed connected smooth complex surface
with vanishing first Chern class and with no global holomorphic 1-form.
A K3 surface is a 4-dimensional real manifold. Kodaira [166] observed that
each K3 surface is diffeomorphic to a unique simply-connected orientable manifold,
the quartic hypersurface
X^4 = {[zo: Z1: Z2: Z3] E <CJP'^3 : tz; = 0} C <CJP'^3.
J=OSiu proved [384] that every K3 admits some K ahler metric. Furthermore, Yau's
proof [447] of the Calabi conjecture showed that each Kahler class of a K3 surface
contains a unique Ricci-flat K ahler metric.
We have the following result (see [128] for an exposition).
PROPOSITION 35.7. Any Kahler-Einstein K3 surface (M^4 , g) is linearly sta-
ble for the Ricci-De Turck flow. That is, its Lichnerowicz Laplacian is negative
semidefinite. The kernel is
c: (g) EB G,
where c: (g) is the product of the 3-dimensional space of parallel self-dual 2 -forms
and the 19-dimensional space of harmonic anti-self-dual 2-forms.
1.5. Manifolds that admit nonzero parallel spinors.
Work [88] of X. Dai, X. Wang, and G. Wei together with earlier results [430]
of M. Wang allow a substantial generalization of Proposition 35.7:
THEOREM 35.8. If a compact Riemannian manifold (Mn, g) is covered by a spin
manifold that admits nonzero parallel spinors, then 6.L has nonpositive spectrum.
REMARK 35.9. This result generalizes Proposition 35.7 because any manifold
satisfying the hypotheses of Theorem 35.8 is necessarily Ricci flat.
The authors of [88] prove Theorem 35.8 by exhibiting the Lichnerowicz Lapla-
cian (with opposite sign convention) as the square of a twisted Dirac operator.
The idea is to view h as the section of a vector bundle with a differential oper-
ator such that the Lichnerowicz Laplacian is the square of that operator. They
achieve this for spin manifolds with parallel spinors using the Dirac operator: Let
(Mn,g) be a compact spin manifold with spinor bundle 5--+ Mn. If a is a par-
allel nonzero spinor, then (Mn,g) is necessarily Ricci flat. Define a linear map
Cf? : 52 --+ 5 0 T* Mn by