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On the Stability of Riemannian Manifold with Parallel Spinors
Xianzhe Dai∗^ Xiaodong Wang†^ Guofang Wei‡
Dedicated to Jeff Cheeger for his sixtieth birthday
Abstract Inspired by the recent work [HHM03], we prove two stability results for compact Riemannian manifolds with nonzero parallel spinors. Our first result says that Ricci flat metrics which also admit nonzero parallel spinors are stable (in the direction of changes in conformal structures) as the critical points of the total scalar curvature functional. Our second result, which is a local version of the first one, shows that any metric of positive scalar curvature cannot lie too close to a metric with nonzero parallel spinor. We also prove a rigidity result for special holonomy metrics. In the case of SU (m) holonomy, the rigidity result implies that scalar flat deformations of Calabi-Yau metric must be Calabi-Yau. Finally we explore the connection with a positive mass theorem of [D03], which presents another approach to proving these stability and rigidity results.
1 Introduction
One of the most fruitful approaches to finding the ‘best’ (or canonical) metric on a manifold has been through the critical points of a natural geometric functional. In this approach one is led to the study of variational problems and it is important to understand the stability issue associated to the variational problem. Consider the space M of Riemannian metrics on a compact manifold M (our manifolds here are assumed to have empty boundary). It is well known that the critical points of the total scalar curvature functional (also known as the Hilbert-Einstein action in general relativity) are Ricci flat metrics. It is also well known that the total scalar curvature functional behaves in opposite ways along the conformal deformations and its transversal directions (i.e., when the conformal structure changes). The variational problem in the conformal class of a metric (volume normalized) is the famous Yamabe problem, which was resolved by Aubin and Schoen. In this paper we study the stability for the total scalar curvature functional when we restrict to the transversal directions, that is, the space of conformal structures. This has to do with the second variation of the total scalar curvature functional restricted to the traceless transverse symmetric 2-tensors, which is given in terms of Lichnerowicz Laplacian [Bes87]. Our first result shows that Riemannian manifolds with nonzero parallel spinors (which are necessarily Ricci flat) are stable in this sense.
Theorem 1.1 If a compact Riemannian manifold (M, g) has a cover which is spin and admits nonzero parallel spinors, then the Lichnerowicz Laplacian Lg is positive semi-definite.
∗Math Dept, UCSB, Santa Barbara, CA 93106 Email: [email protected]. †Math Dept, MIT, Cambridge, MA 02139, USA. Email:[email protected]. Partially supported by NSF Grant # DMS 0202122 ‡Math Dept. UCSB. Email: [email protected]. Partially supported by NSF Grant # DMS-0204187.
This makes essential use of a Bochner type formula relating the Lichnerowicz Laplacian to the square of a twisted Dirac operator for Riemannian manifolds with nonzero parallel spinors, which is a special case of a result in [Wa91, Proposition 2.4], where the general case of Killing spinor is discussed. Also, we note that the special case of Theorem 1.1 for K3 surfaces is proved in [GIK02]. Theorem 1.1 settles an open question raised in [KW75] about thirty years ago in the case when the Ricci flat manifold has a spin cover with nonzero parallel spinors. It is an interesting open question of how special our metric is compared to the general Ricci flat metric (Cf. Section 5) but we note that, so far, all known examples of compact Ricci flat manifolds are of this type, namely, they admit a spin cover with nonzero parallel spinors. In a very recent work [CHI04], Cao-Hamilton-Ilmanen studied the stability problem for Ricci solitons and Ricci shrinking solitons using the functionals introduced by Perelman [P02] and showed that they are governed by the Lichnerowicz Laplacian. Thus, as an application of Theorem 1.1, Cao-Hamilton-Ilmanen deduce that compact manifolds with nonzero parallel spinors are also stable as Ricci soliton [CHI04]. We next prove that, in fact, there exists a neighborhood of the metric with nonzero parallel spinors, which contains no metrics with positive scalar curvature. This can be thought of as a local version of our previous (infinitesimal) stability result.
Theorem 1.2 Let (M, g) be a compact Riemannian manifold which admits a spin cover with nonzero parallel spinors. Then g cannot be deformed to positive scalar curvature metrics. By this we mean that there exists no path of metrics gt such that g 0 = g and the scalar curvature S(gt) > 0 for t > 0. In fact, if (M, g) is simply connected, then there is a neighborhood of g in the space of metrics which does not contain any metrics with positive scalar curvature.
The existence of metrics with positive scalar curvature is a well studied subject, with important work such as [L63], [H74], [SY79-2], [GL80], culminating in the solution of the Gromov-Lawson conjecture in [S92] for simply connected manifolds. Thus, a K3 surface does not admit any metric of positive scalar curvature, but a simply connected Calabi-Yau 3-fold does. Of course, a Calabi-Yau admits nonzero parallel spinors (in fact, having nonzero parallel spinors is more or less equivalent to having special holonomy except quaternionic K¨ahler, cf. Section 3). Thus on a Calabi-Yau 3-fold there exist metrics of positive scalar curvature but they cannot be too close to the Calabi-Yau metric. One should also contrast our result with an old result of Bourguignon, which says that a metric with zero scalar curvature but nonzero Ricci curvature can always be deformed to a metric with positive scalar curvature (essentially by Ricci flow). The local stability theorem implies a rigidity result, Theorem 3.4, from which we deduce the following interesting application.
Theorem 1.3 Any scalar flat deformation of a Calabi-Yau metric on a compact manifold must be Calabi-Yau. In fact, any deformation with nonnegative scalar curvature of a Calabi-Yau metric on a compact manifold is necessarily a Calabi-Yau deformation. The same is true for the other special holonomy metrics, i.e., hyperk¨ahler, G 2 , and Spin(7).
Our result generalized a theorem in [Wa91] for Einstein deformations. The proof of our second result, the local stability theorem, actually gives a very nice picture of what happens to scalar curvature near a metric with parallel spinor (we’ll call it special holonomy metric): one has the finite dimensional moduli of special holonomy metrics; along the normal direc- tions, the scalar curvature of the Yamabe metric in the conformal class will go negative. In other words, we have
Theorem 1.4 The Calabi-Yau (and other special holonomy) metrics are local maxima for the Yam- abe invariant.
the first eigenfunction, normalized to satisfy
M ψg^ dVg^ = 1, i.e.
∆g ψg + cnSg ψg = λ(g)ψg , (2.1) ∫
M
ψg dVg = 1 (2.2)
So defined, ψg is then uniquely determined and is in fact positive. The functional λ(g), studied first by Kazdan and Warner [KW75], has some nice properties. Though it is not conformally invariant, its sign is conformally invariant. By the solution of the Yamabe problem, any metric g can be conformally deformed to have constant scalar curvature. The sign of the constant is the same as that of λ(g). In fact the metric ψ^4 g/ (n−2)g has scalar curvature
cnλ(g)ψ− g 4 /(n−2)whose sign is determined by λ(g). Obviously λ(g 0 ) = 0 and the corresponding eigenfunction is ψ 0 ≡ 1. We present the variational analysis of the functional λ, essentially following [KW75] with some modification and simplification. (Our notations are also slightly different). We first indicate that λ(g) and ψg are smooth in g near g 0. Let U be the space of metrics on M , V = {ψ ∈ C∞(M )|
M ψdVg^0 = 1}^ and^ W^ =^ {ψ^ ∈^ C
∞(M )| ∫
M ψdVg^0 = 0}.^ We define F : U × V → W as follows
F (g, ψ) =
[
∆g ψg^ + cnSg ψg^ − cn
Sg ψg^ dVg
ψg
]
dVg dVg 0
where ψg^ = ψ dVg 0 dVg. Note^
ψg^ dVg = 1. Obviously F (g, ψ) = 0 iff ψg^ is an eigenfunction of ∆g + cnSg
and is the first eigenfunction iff ψ > 0. We have F (g 0 , ψ 0 ) = 0 and the linearization in the second variable at (g 0 , ψ 0 ) is easily seen to be ∆g 0 : W → W. Since this is an isomorphism, we conclude by the implicit function theorem that λ(g) and ψg are smoothly dependent on g in a neighborhood of g 0. Let g(t) for t ∈ (−, ) be a smooth family of metrics with g(0) = g 0. Before we analyze the variation of λ near g 0 , we collect a few formulas:
Ric = ˙^1 2
(∇∗∇h − 2
◦ Rh) − δ∗δh −
D^2 tr h + Ric ◦ h, (2.4)
S˙ = −〈h, Ric 〉 + δ^2 h + ∆tr h, (2.5)
∆˙f = 〈h, D^2 f 〉 − 〈δh +^1 2
dtr h, df 〉. (2.6)
where (
◦ Rh)ij = Rikjlhkl denotes the action of the curvature on symmetric 2-tensors, D^2 denotes the Hessian, and k ◦ h denotes the symmetric 2-tensor associated to the composition of k and h viewed as (1, 1)-tensors via the metric, i.e., as linear maps from T M to itself. An “upperdot” denotes the derivative with respect to t and h = ˙g. Differentiating (2.1) in t gives
λψ˙ = ∆ ψ˙ + ∆˙ψ + cn( Sψ˙ + S ψ˙) − λ ψ˙ (2.7)
We integrate over M and compute using the above formulas and integration by parts
λ˙ =
M
∆^ ˙ψ + cn( Sψ˙ + S ψ˙) − λ
M
ψ^ ˙
M
〈h, D^2 ψ〉 − 〈δh +
dtr h, dψ〉 + cn[−〈h, Ric 〉ψ + δ^2 hψ + ∆tr hψ + S ψ˙] +
λ 2
M
ψtr h
= cn
M
〈h, −ψRic + D^2 ψ〉 −
n n − 2
tr h∆ψ + S ψ˙ +
λ 2
M
ψtr h
Therefore the first variation formula is
λ˙ = cn
M
〈h, −ψRic + D^2 ψ − n n − 2
∆ψg〉 + S ψ˙ + λ 2
M
ψtr h. (2.8)
As g 0 is scalar flat, we have λ(0) = 0, ψ 0 = 1 and hence the elegant
λ˙(0) = −cn
M
〈Ric (g 0 ), h〉dVg 0. (2.9)
This shows that g 0 is a critical point of λ iff it is Ricci flat. Remark One can analyze the variation of the first eigenvalue of ∆g + cSg for any constant c in the same fashion. It turns out that it has the most elegant first variation for general metric exactly for c = 14 , which corresponds to the Perelman’s λ-functional [P02]. See also [CHI04]. As a corollary we have
Proposition 2.1 (Bourguignon) If g 0 has zero scalar curvature but non-zero Ricci curvature, then it can be deformed to a metric of positive scalar curvature.
Proof: Take h = −Ric (g 0 ) and g(t) = g 0 + th. Then λ˙(0) = cn
M |Ric (g^0 )|
(^2) dVg 0 >^ 0 and hence λ(g(t)) > 0 for t > 0 small. Then ψ^4 t /(n−2)g(t) has positive scalar curvature with ψt being the positive first eigenfunction of g(t).
Now, a natural question is then, what happens if Ric (g 0 ) is identically zero. This is exactly the question discussed by Hertog, Horowitz and Maeda [HHM03] who, based on supersymmetry, argued that in a neighborhood of a Calabi-Yau metric there is no metric of positive scalar curvature. For this purpose we need to derive the second variation for λ. We now assume that g 0 is Ricci flat. Differentiating (2.8) at t = 0 and using (2.4) and (2.5) we get
¨λ(0) = cn
M
〈h, − Ric +˙ D^2 ψ˙ − n n − 2
∆ ψg˙ 0 〉 + S˙ ψ˙
= cn
M
〈h, −
(∇∗∇h − 2
◦ Rh) + δ∗δh +
D^2 tr h + 2D^2 ψ˙ −
n − 2
∆ ψg˙ 0 〉
def = cn
M
〈h, F(h)〉.
And ψ˙ at t = 0 appearing in the formula is determined by the equation
∆ ψ˙ = −cn S˙ = −cn(δ^2 h + ∆tr h). (2.10)
The symmetric tensor h can be decomposed as h = ¯h + h 1 + h 2 , where h 1 = LX g 0 for some vector field X, h 2 = ug 0 for some smooth function u and ¯h is transverse traceless, that is tr ¯h = 0 and δ¯h = 0. Let φt be the flow generated by the vector field X. Let g(t) = φ∗ t g 0 and note obviously its variation is h 1 = LX g 0. Then Ric (gt) = φ∗ t Ric g 0 = 0. Differentiating in t and using (2.4) we have
(∇∗∇h 1 − 2
◦ Rh 1 ) + δ∗δh 1 +
D^2 tr h 1 = 0. (2.11)
Similarly by working with the scalar curvature we have
δ^2 h 1 + ∆tr h 1 = 0 (2.12)
If M is a Riemannian manifold, then for the Levi-Civita connection on T M , we have R(X, Y, Z, W ) = 〈RXY Z, W 〉. We often work with an orthonormal frame {e 1 ,... , en} and its dual frame {e^1 ,... , en}. Set Rijkl = R(ei, ej , ek, el). The spinor bundle has a natural connection induced by the Levi-Civita connection on T M. For a spinor σ, we have
RXY σ =
R(X, Y, ei, ej )eiej · σ. (2.18)
If σ 0 6 = 0 is a parallel spinor, then Rklij eiej · σ 0 = 0. (2.19)
It is well known this implies Ric = 0 by computation. From now on, we assume M has a parallel spinor σ 0 6 = 0, which, without loss of generality, is normalized to be of unit length. We define a linear map Φ : S^2 (M ) → S ⊗ T ∗M by
Φ(h) = hij ei · σ 0 ⊗ ej^. (2.20)
It is easy to check that the definition is independent of the choice of the orthonormal frame {e 1 ,... , en}.
Lemma 2.3 The map Φ satisfies the following properties:
- Re 〈Φ(h), Φ(˜h)〉 = 〈h, ˜h〉,
- ∇X Φ(h) = Φ(∇X h).
Proof: We compute
〈Φ(h), Φ(˜h)〉 = hij ˜hkl〈ei · σ 0 ⊗ ej^ , ek · σ 0 ⊗ el〉 = hil˜hkl〈ei · σ 0 , ek · σ 0 〉 = −hilh˜kl〈σ 0 , eiek · σ 0 〉 = hkl˜hkl −
i 6 =k
hil˜hkl〈σ 0 , eiek · σ 0 〉
= 〈h, ˜h〉 −
i 6 =k
hil˜hkl〈σ 0 , eiek · σ 0 〉.
Now, for i 6 = k,
〈σ 0 , eiek · σ 0 〉 = 〈ekei · σ 0 , σ 0 〉 = −〈eiek · σ 0 , σ 0 〉 = −〈σ 0 , eiek · σ 0 〉.
That is, 〈σ 0 , eiek · σ 0 〉 is purely imaginary. Taking the real part of the previous equation proves the first assertion. To prove the second one, we choose our orthonormal frame such that ∇ei = 0 at p and compute at p
∇X Φ(h) = Xhij ei · σ 0 ⊗ ej = ∇X h(ei, ej )ei · σ 0 ⊗ ej = Φ(∇X h).
The following interesting Bochner type formula, which follows from [Wa91, Proposition 2.4], plays an important role here.
Proposition 2.4 (M. Wang) Let h be a symmetric 2 -tensor on M. Then
D∗DΦ(h) = Φ(∇∗∇h − 2
◦ Rh). (2.21)
Moreover, Lg h = 0 iff DΦ(h) = 0.
Proof: We present a lightly different proof here. Choose an orthonormal frame {e 1 ,... , en} near a point p such that ∇ei = 0 at p. We compute at p
D∗DΦ(h) = ∇ek ∇el h(ei, ej )ekelei · σ 0 ⊗ ej
= −∇ek ∇ek h(ei, ej )ei · σ 0 ⊗ ej^ −
Rek el h(ei, ej )ekelei · σ 0 ⊗ ej
= Φ(∇∗∇h) +
Rkljphipekelei · σ 0 ⊗ ej^ +
Rkliphpj ekelei · σ 0 ⊗ ej^.
By using twice the Clifford relation eiej + ej ei = − 2 δij we have
1 2
Rkljphipekelei · σ 0 =
Rkljphipeiekel · σ 0 + Rkljphkpel · σ 0 − Rkljphlpek · σ 0
= −2(
◦ Rh)kj ek · σ 0 ,
where in the last equality we used Rkljpekel · σ 0 = 0 by (2.19). Similarly (in fact easier) one can show using also the fact Ric = 0 1 2 Rkliphpj ekelei · σ 0 = 0.
Thus we get
D∗DΦ(h) = Φ(∇∗∇h − 2
◦ Rh). By Lemma 2.3, Φ preserves the metrics. Hence, Lg h = 0 iff DΦ(h) = 0. By using Lemma 2.3, Proposition 2.4, and working on a covering space, we obtain
Theorem 2.5 If a compact Riemannian manifold (M, g) has a cover which is spin and admits nonzero parallel spinors, then the Lichnerowicz Laplacian Lg is positive semi-definite.
Proof: Let π : ( M ,ˆ ˆg) → (M, g) be the cover. Clearly the following diagram commutes
Lg : S^2 (M ) → S^2 (M ) π∗^ ↓ π∗^ ↓ Lˆg : S^2 ( Mˆ ) → S^2 ( Mˆ ).
Now if we denote by 〈· , ·〉 the pointwise inner product on symmetric 2-tensors and (· , ·) the L^2 inner product, i.e., for example,
(h, h′)g =
M
〈h, h′〉g dvol(g),
then we have
〈Lg h, h〉g = 〈Lˆg π∗(h), π∗(h)〉ˆg. (2.22)
Thus, for a fundamental domain F of M in Mˆ , one has
For this purpose, we need to understand the geometry of (M, g 0 ) better. According to [Wa89](cf. [J00] 3.6), if (M, g 0 ) is a compact, simply connected, irreducible Riemannian spin manifold of di- mension n with a parallel spinor, then one of the following holds
- n = 2m, m ≥ 2, the holonomy group is SU (m),
- n = 4m, m ≥ 2, the holonomy group is Sp(m),
- n = 8, the holonomy group is Spin(7),
- n = 7, the holonomy group is G 2.
In cases 2 and 3, it is further shown in [Wa89] that the index of the Dirac operator is nonzero, hence by Lichnerowicz’s theorem there is no metric of positive scalar curvature. Therefore the theorem is “trivial” except in cases 1 and 4. Suppose (M, g 0 ) is a compact Riemannian manifold of dimension n = 2m with holonomy SU (m). This is a Calabi-Yau manifold. By Yau’s solution of Calabi conjecture [Y77, Y78] and the theorem of Bogomolov-Tian-Todorov [Bo78], [T86], [To89], the universal deformation space Σ of Calabi-Yau metrics is smooth of dimension h^1 ,^1 + 2hm−^1 ,^1 − 1 (it is one less than the usual number because we normalize the volume and hence discount the trivial deformation of scaling). Its tangent space at g 0 must be a subspace of Wg 0 for Wg 0 is the Zariski tangent space of the moduli space of Ricci flat metrics. In fact we have
Lemma 3.2 Tg 0 Σ = Wg 0.
Proof: This follows from a theorem of Koiso [Ko80] which says Einstein deformations of a K¨ahler- Einstein metric are also K¨ahler, provided that first Chern class is nonpositive and the complex deformation are unobstructed, which is guaranteed by Bogomolov-Tian-Todorov theorem [T86], [To89]. It can also be easily seen from our approach. For a Calabi-Yau manifold its spinor bundle is
S+(M ) =
k even
∧^0 ,k(M ), S−(M ) =
k odd
∧^0 ,k(M ). (3.2)
The Clifford action at a point p ∈ M is defined by
X · α =
2(π^0 ,^1 (X∗) ∧ α − π^0 ,^1 (X)yα) (3.3)
for any X ∈ TpM and α ∈ Sp(M ) and the parallel spinor σ 0 ∈ C∞(S+(M )) can be taken as the function which is identically 1. Let J be the complex structure. Then we have Wg 0 = W +^ ⊕ W −, where
W +^ = {h ∈ Wg 0 |h(J, J) = h}, W −^ = {h ∈ Wg 0 |h(J, J) = −h} (3.4)
We choose a local orthonormal (1, 0) frame {X 1 ,... , Xm} for T 1 ,^0 M and its dual frame {θ^1 ,... , θm}. By straightforward computation we have for h ∈ W +
Φ(h) = h( X¯i, Xj )θ¯i^ ⊗ θj^ (3.5)
which can be identified with the real (1, 1) form
− 1 h( X¯i, Xj )θ¯i^ ∧ θj^. The Dirac operator is then identified as
∗ ) (cf. Morgan [M96]). Therefore W +^ is identified with the space of harmonic (1, 1)-forms orthogonal to the K¨ahler form ω. Similarly W −^ can be identified as H^1 (M, Θ)−H^0 ,^2 (M ), where Θ is the holomorphic tangent bundle. As H^0 ,^2 (M ) = 0 and H^1 (M, Θ) ∼= Hm−^1 ,^1 (M ) by the Hodge theory, we have dim Wg 0 = h^1 ,^1 + 2hm−^1 ,^1 − 1. This is exactly the dimension of the moduli space of Calabi-Yau metrics.
We now turn to the proof of our local stability theorem in the case of Calabi-Yau manifold. Let M be the space of Riemannian metric of volume 1. By Ebin’s slice theorem, there is a real submanifold S containing g 0 , which is a slice for the action of the diffeomorphism group on M. The tangent space
Tg 0 S = {h|δg 0 h = 0,
M
tr (^) g 0 hdVg 0 = 0.} (3.6)
Let C ⊂ S be the submanifold of constant scalar curvatures metrics.If g ∈ M is a metric of positive scalar curvature very close to g 0 , then by the solution of the Yamabe problem there is a metric ˜g ∈ C conformal to g and with constant positive scalar curvature. Moreover as g is close to g 0 which is the unique Yamabe solution in its conformal class, ˜g is also close to g 0. Therefore to prove the theorem, it suffices to work on C. It is easy to see
Tg 0 C = {h|δg 0 h = 0, tr (^) g 0 h = 0.} (3.7)
It contains the finite dimensional submanifold of Calabi-Yau metrics E. We now restrict our function λ to C. It is identically zero on E. Moreover, by Lemma 3.2 and Proposition 2.2, D^2 λ is negative definite on the normal bundle. Therefore there is a possibly smaller neighborhood of E ⊂ C, still denoted by U, such that λ is negative on U − E.
Next we consider the case 4, that is, the case of G 2 manifold. Our basic references are Bryant [B89, B03] and Joyce [J00]. Let (M, g 0 ) be a compact Riemannian manifold with holonomy group G 2. We denote the fundamental 3-form by φ. With a local G 2 -frame {e 1 , e 2 ,... , e 7 } and the dual frame {e^1 , e^2 ,... , e^7 } we have
φ = e^123 + e^145 + e^167 + e^246 − e^257 − e^347 − e^356 , (3.8) ∗φ = e^4567 + e^2367 + e^2345 + e^1357 − e^1346 − e^1256 − e^1247. (3.9)
We also define the cross product P : T M × T M → T M by
〈P (X, Y ), Z〉 = φ(X, Y, Z). (3.10)
The cross product has many wonderful properties. We list what we need in the following lemma.
Lemma 3.3 For any tangent vectors X, Y, Z
- P (X, Y ) = −P (Y, X).
- 〈P (X, Y ), P (X, Z)〉 = |X|^2 〈Y, Z〉 − 〈X, Y 〉〈X, Z〉,
- P (X, P (X, Y )) = −|X|^2 Y + 〈X, Y 〉X,
- Xy(Y y ∗ φ) = −P (X, Y )yφ + X∗^ ∧ Y ∗.
Proof: The first three identities are proved in Bryant [B89]. The fourth can be proven by the same idea: it is obviously true for X = e 1 , Y = e 2 and the general case follows by the transitivity of G 2 on orthonormal pairs.
The spinor bundle is S(M ) = R ⊕ T M with the first factor being the trivial line bundle. The Clifford action at p ∈ M is defined by
X · (a, Y ) = (−〈X, Y 〉, aX + P (X, Y )) (3.11)
for any X, Y ∈ TpM. The parallel spinor σ 0 = (1, 0). One easily check φ(X, Y, Z) = −〈X · Y · Z · σ 0 , σ 0 〉. It is also obvious that S(M ) ⊗ T ∗M = T ∗M ⊕ (T M ⊗ T ∗M ) and for any symmetric 2-tensor
Φ(h) = (0, hij ei ⊗ ej^ ) (3.12)
where in the last step we used tr h = 0 and δh = 0. Then by Lemma 3.3 we continue
∗dΨ(h) = −hij,kej^ ∧ (eky(eiy ∗ φ)) = −hij,kej^ ∧ (P (ei, ek)yφ) + hij,kej^ ∧ ei^ ∧ ek = 0,
where in the last step we used (3.22) and the fact that h is symmetric. Therefore for any h ∈ Wg 0 , the 3-form Ψ(h) is harmonic. This proves Wg 0 = Vg 0 is the tangent space to the moduli space of G 2 metrics. The rest of argument is the same as in the Calabi-Yau case. By the above argument we also obtain the following rigidity theorem which generalize the Einstein deformation result, Theorem 3.1, of [Wa91].
Theorem 3.4 Let (M, g 0 ) be a compact, simply connected Riemannian spin manifold of dimension n with a parallel spinor. Then there exists a neighborhood U of g 0 in the space of smooth Riemannian metrics on M such that any metric with nonnegative scalar curvature in U must in fact admit a parallel spinor (and hence Ricci flat in particular).
Proof: We give the proof in the Calabi-Yau case. As shown in the proof of Theorem 3.1 λ ≤ 0 on U and is negative on U − E, where E is the moduli space of Calabi-Yau metrics. If g ∈ U has nonnegative scalar curvature, then λ(g) ≥ 0. Therefore λ(g) = 0 and g ∈ E.
Note that our proof actually gives a very nice picture of what happens to scalar curvature near a metric with a parallel spinor (let’s call it special holonomy metric): one has the finite dimensional smooth moduli of special holonomy metrics; along the normal directions, the scalar curvature of the Yamabe metric in the conformal class will go negative. That is, we have
Theorem 3.5 Let (M, g 0 ) be a compact, simply connected Riemannian spin manifold of dimension n with a parallel spinor. Then g 0 is a local maximum of the Yamabe invariant.
Proof: Recall that the Yamabe invariant of a metric g is
μ(g) = inf f ∈ C∞(M ), f > 0
M (^
n− 1 n− 2 |df^ |
2 g +^ S(g)f^ (^2) )dvol(g) (
M f^
2 n/(n−2)dvol(g))(n−2)/n (3.23)
and it is a conformal invariant. The corresponding Euler-Lagrange equation is
4
n − 1 n − 2 ∆g f + S(g)f = μ(g)f (n+2)/(n−2). (3.24)
Its nontrivial solution, whose existence guaranteed by the solution of Yamabe problem, gives rise to the so-called Yamabe metric f 4 /(n−2)g which has constant scalar curvature μ(g). Note that the left hand side of (3.24) is (up to the positive multiple of 4 n n−−^12 ) the conformal Laplacian and the numerator of the quotient in (3.23) is the quadratic form defined by the conformal Laplacian (again up to the positive multiple of 4 n n−−^12 ). Thus, μ(g) < 0 if the first eigenvalue of the conformal Laplacian λ(g) < 0. Now μ(g 0 ) = λ(g) = 0 since g 0 is scalar flat. Since we have shown that g 0 is a local maximum of λ(g), our result follows.
4 A uniform approach: connection with the positive mass
theorem
In this section we explore a remarkable connection between the stability of the Riemannian manifold M and the positive mass theorem on R^3 ×M. It provides us with a uniform approach to the stability
problem, rather than the case by case treatment of the previous section. The original positive mass theorem [SY79-1], [Wi81] is related to the stability of the Minkowski space as the vacuum (or minimal energy state). In superstring theory this vacuum is replaced by the product of the Minkowski space with a Calabi-Yau manifold [CHSW85]. The positive mass theorem for spaces which are asymptotic to Rk^ × M at infinity is proved in [D03]. The following result is a special case of what is considered in [D03].
Theorem 4.1 Let M be a compact Riemannian spin manifold with nonzero parallel spinors. If ˜g is a complete Riemannian metric on R^3 × M which is asymptotic of order > 1 / 2 to the product metric at infinity and with nonnegative scalar curvature, then its mass m(˜g) ≥ 0. (Moreover, m(˜g) = 0 iff ˜g is isometric to the product metric.)
Remark The result of [D03] is stated for simply connected compact Riemannian spin manifolds M with special holonomy with the exception of quaternionic K¨ahler but the proof only uses the existence of a nonzero parallel spinor, which is equivalent to the holonomy condition by a result of [Wa89]. Also, the simply connectedness of M is not needed here since the total space is the product R^3 × M. Using the positive mass theorem, we prove the following deformation stability for compact Rie- mannian spin manifold with nonzero parallel spinors. This is a special case of our previous local stability theorem. However, as we point out earlier, this approach treats all cases of special holonomy manifolds at the same time.
Theorem 4.2 Let M be a compact spin manifold and g a Riemannian metric which admits nonzero parallel spinors. Then g cannot be deformed to metrics with positive scalar curvature. Namely, there is no path of metrics gs such that g 0 = g and S(gs) > 0 for s > 0.
Remark The condition on the deformation can be relaxed to S(gs) ≥ 0 and S(gsi ) > 0 for a sequence si → 0. This has the following interesting consequence, which, once again, is a special case of the rigidity result of the previous section, Theorem 3.4.
Theorem 4.3 Any scalar flat deformation of Calabi-Yau metric on a compact manifold must also be Calabi-Yau. Indeed, any deformation of Calabi-Yau metric on a compact manifold with nonnegative scalar curvature must be a Calabi-Yau deformation.
Proof: Let (M, g) be a compact Calabi-Yau manifold and gs be a scalar flat deformation of g, i.e., S(gs) = 0 for all s. We will show by Theorem 4.2 that gs must also be Ricci flat. It follows then by a theorem of Koiso and the Bogomolov-Tian-Todorov theorem, gs must be in fact Calabi-Yau. We first show that for s sufficiently small, gs must be Ricci flat. If not, there is a sequence si → 0 such that Ric(gsi ) 6 = 0. Hence this will also be true in a small neighborhood of si. By Bourguignon’s theorem, we can deform gs slightly in such neighborhood so that it will now have positive scalar curvature. Moreover we can do this while keeping the metrics gs unchanged outside this neighborhood. This shows that we have a path of metrics satisfying the conditions in the Remark above. Hence contradictory to Theorem 4.2. Once we know that gs is Ricci flat for s sufficiently small, we then deduce that gs is Calabi-Yau for s sufficiently small, by using Koiso’s Theorem [Bes87], [Ko80] and Bogomolov-Tian-Todorov Theorem [T86], [Bo78], [To89]. Then the above argument applies again to extend to the whole deformation. The case of deformation with nonnegative scalar curvature is dealt with similarly.
To prove Theorem 4.2, we show that if there is such a deformation of g, then one can construct a complete Riemannian metric ˜g on R^3 × M which is asymptotic of order 1 to the product metric at
The curvature tensors involving the U 0 direction can be computed from the metric:
〈 R˜(U 0 , Ui)U 0 , Ui〉 = − m r
〈 R˜(U 0 , Yα)U 0 , Yβ 〉 =
m′r − m r^2 gαβ,r −
2 m(r) r
s,l
d dr
[
gαs,r gls
]
glβ
2 m(r) r
s,l
gαs,r glsglβ,r. (4.9)
Therefore, using (4.8) (4.9),
R˜ 00 = 2
m r^3
m′ r^2
m′r − m r^2
g′^ −
2 m(r) r
g′′
2 m(r) r
α,β,s,l
gαs,r glβ,r glsgαβ^. (4.10)
Similarly by (4.8) (4.5) (4.6) and (4.9) (4.6) (4.7)
R˜ii = − m r
2 m r
r 2
2 m(r) r
g′, (4.11)
R˜αβ = 1 2
m′r − m r^2 gαβ,r −
2 m(r) r
s,l
d dr
[
gαs,r gls
]
glβ
2 m(r) r
s,l
gαβ,r gsl,r gls^ −
r
2 m(r) r
gαβ,r + Rαβ (M, g(r)) (4.12)
Finally, using (4.10) (4.11) (4.12), we have
S˜ = SM (g(r)) + m′
r^2
g′ r
m r^2
g′^ −
2 m(r) r
) [
g′′^ +
r
g′^ +
(g′)^2 −
∂r gαβ ∂r gαβ
]
We now turn to the construction of asymptotically flat metrics of the type (4.1) with nonnegative scalar curvature and negative mass.
Proposition 4.6 Let gs, s ∈ [0, 1], be a one-parameter family of metrics on M such that
SM (g 1 ) ≥ a 0 > 0 , SM (g 0 ) ≥ 0.
If in addition
max |∂s(gs)αβ ∂sgαβs | ≤
, max |g′ s| ≤
, max |g s′′ | ≤
(here we did not try to get the optimal constants), and
S− M (gs) def = max(0, −SM (gs)) ≤
a 0 10
for all s ∈ [0, 1], then there exists m(r) with m(0) = m′(0) = m′′(0) = 0, limr→∞ m(r) = m∞ < 0 and g(r) with g(r) = g 0 for r sufficiently large, such that the metric ˜g in (4.1) has scalar curvature S^ ˜ ≥ 0.
Proof: We note first that by reparametrizing the interval [0, 1] we can arrange the family gs so that gs = g 0 for s sufficiently small and gs = g 1 for s sufficiently close to 1 and the assumption on gs still holds (with different constants). This will ensure that the following gluing construction will produce a smooth metric. Now, the function m(r) and the metrics g(r) will be constructed separately on the intervals [0, r 1 ], [r 1 , r 2 ], [r 2 , r 3 ] and [r 3 , ∞), with r 1 , r 2 , r 3 to be chosen appropriately. First of all, define g(r) = g 1 for 0 ≤ r ≤ r 2. (4.14)
For the function m(r), we let m(r) = − a 120 r^3 when 0 ≤ r ≤ r 1 and just require that m′(r) ≥ − a 40 r^2 for r 1 ≤ r ≤ r 2 , m(r 2 ) = m(r 1 ) and m′(r 2 ) = −m′(r 1 ) = a 40 r^21. We note that there are many such choices for m(r) on the interval [r 1 , r 2 ] and further r 2 can be taken arbitrarily close to r 1. For definiteness we put r 2 = r 1 + 1. In the region 0 ≤ r ≤ r 2 , since the metrics g(r) = g 1 does not change with r, one easily sees from the scalar curvature formula, Lemma 4.5, that
S˜ = SM (g 1 ) + m′(r)^4 r^2
For the interval [r 2 , r 3 ], we define m(r) to be the linear function
m(r) = m(r 2 ) +
a 0 4 r^21 (r − r 2 )
and g(r) = g r 3 −r r 3 −r 2
Note that by our remark at the beginning of the proof, g(r) glues smoothly in r at r 2 (similar remark applies to r 3 below). To make sure that the scalar curvature S˜ of the total space remains nonnegative in this region, we need to choose the parameters r 1 , r 3 appropriately. Let C 1 = max |∂s(gs)αβ ∂sgαβs |, C 2 = max |g′ s|, C 3 = max |g s′′ |. Then we have
S˜ ≥ SM (g(r)) + a^0 4
r^21
r^2
C 2
r(r 3 − r 2 )
a 0 12
r^21 r^2
|−r 1 + 3(r − r 2 )|
C 2
r 3 − r 2
−
a 0 6
r 12 r
| − r 1 + 3(r − r 2 )|
C 1
4(r 3 − r 2 )^2
C 3
(r 3 − r 2 )^2
2 C 2
r(r 3 − r 2 )
C 22
4(r 3 − r 2 )^2
This yields
S˜ ≥ −S− M (gs) + a^0 4
r^21 r^2
(4 − A(r)) −
(r 3 − r 2 )^2
B(r), (4.16)
where
A(r) = C 2 r r 3 − r 2
5 C 2
4 C 3 + C 1 + C 22
r r 3 − r 2
| − r 1 + 3(r − r 2 )| r 3 − r 2
and
B(r) =
C 1 + 4C 3 + C 22
+ 2C 2
r 3 − r 2 r
We now set r 3 = r 2 + 17 r 1. By the given bounds on C 1 , C 2 , C 3 we have, for r 2 ≤ r ≤ r 3 , r 1 ≥ 7,
|A(r)| ≤ 3 , |B(r)| ≤ 1
We then take r 1 sufficiently large so that
1 (r 3 − r 2 )^2
a 0 4
)^2 ,
It is an interesting open question of what, if any, restriction the Ricci flat condition will impose on the holonomy. All know examples of simply connected compact Ricci flat manifolds are of special holonomy and hence admit nonzero parallel spinors. From our local stability theorem, scalar flat metrics sufficiently close to a Calabi-Yau metric are necessarily Calabi-Yau themselves. One is thus led to the question of whether there are any scalar flat metrics on a compact Calabi-Yau manifold which are not Calabi-Yau.
Proposition 5.2 If (M, g) is a compact simply connected G 2 manifold or Calabi-Yau n-fold (i.e. dimC = n) with n = 2k + 1, then there are scalar flat metrics on M which are not G 2 or Calabi-Yau, respectively.
Proof: By the theorem of Stolz [S92], M carries a metric of positive scalar curvature (because of the dimension in the case of G 2 manifold, and Aˆ = 0 in the case of Calabi-Yau [Wa89], [J00]). Let g′^ be such a metric. Then, its Yamabe invariant μ(g′) > 0. On the other hand, every compact manifold carries a metric of constant negative scalar curvature [Bes87], say g′′. Then μ(g′′) < 0. Now consider gt = tg′^ + (1 − t)g′′. Since the Yamabe invariant depends continuously on the metric [Bes87], there is a t 0 ∈ (0, 1) such that μ(gt 0 ) = 0 and μ(gt) > 0 for 0 ≤ t < t 0. The Yamabe metric in the conformal class of gt 0 is scalar flat but cannot be a G 2 (Calabi-Yau, resp.) by Theorem 3.5.
Remark For the other special holonomy metric (including the Calabi-Yau SU (2k)), the index Aˆ 6 = 0 [Wa89], [J00]. Thus there is no positive scalar curvature metric on such manifold. Scalar flat metrics on such manifolds are classified by Futaki [Fu93]: they are all products of the special holonomy metrics (with nonzero index). One is also led naturally to the following question: are there any scalar flat but not Ricci flat metrics on a compact Calabi-Yau (or G 2 ) manifold? A more interesting and closely related question is:
Question: Is any Ricci flat metric on a compact Calabi-Yau manifold necessarily K¨ahler (and hence Calabi-Yau)?
We note that the deformation analogue of this question has a positive answer by the work of Koiso, Bogomolov, Tian, Todorov. Moreover, in (real) dimension four, the answer is positive by the work of Hitchin [H74.2] on the rigidity case of Hitchin-Thorpe inequality.
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