The Cauchy problems for Einstein metrics and parallel spinors
aa r X i v : . [ m a t h . DG ] A p r THE CAUCHY PROBLEMS FOR EINSTEIN METRICS ANDPARALLEL SPINORS
BERND AMMANN, ANDREI MOROIANU, AND SERGIU MOROIANU
Abstract.
The restriction of a parallel spinor on some spin manifold Z to a hypersurface M ⊂ Z is a generalized Killing spinor on M . We show, conversely, that in the realanalytic category, every spin manifold ( M, g ) carrying a generalized Killing spinor ψ canbe isometrically embedded as a hypersurface in a spin manifold carrying a parallel spinorwhose restriction to M is ψ . We also answer negatively the corresponding question in thesmooth category. Introduction
This paper aims to solve the problem of extending a spinor from a hypersurface to a par-allel spinor on the total space. This problem is related to that of extending a Riemannianmetric on a hypersurface to an Einstein metric on the total space, since parallel spinorscan only exist over Ricci-flat manifolds.
The Cauchy problem for Einstein metrics.
In the Lorentzian setting, Ricci-flat ormore generally Einstein metrics form the central objects of general relativity. Given aspace-like hypersurface, a Riemannian metric, and a symmetric tensor which plays therole of the second fundamental form, there always exists a local extension to a LorentzianEinstein metric [29], [25], provided that the local conditions given by the contracted Gaussand Codazzi-Mainardi equations are satisfied. One crucial step in the proof is the reductionto an evolution equation which is (weakly) hyperbolic due to the signature of the metric.The corresponding equations in the Riemannian setting are (weakly) elliptic and no generallocal existence results are available.In fact, if (
M, g ) is any hypersurface of an Einstein manifold ( Z , g Z ), then the Weingartentensor W is a symmetric endomorphism field on M which satisfies certain constraints (see(2.9)–(2.10) below, which are contractions of the Gauss and Codazzi-Mainardi equations).Conversely, one can ask the following question: (Q1): If W is a symmetric endomorphism field on M which satisfies thesystem (2.9) – (2.10) , does there exist a isometric embedding of M into a(Riemannian) Einstein manifold ( Z n +1 , g Z ) with Weingarten tensor W ? Is g Z unique near M up to isometry? Date : September 10, 2018.2010
Mathematics Subject Classification.
Key words and phrases.
Cauchy problem, parallel spinors, generalized Killing spinors, Einstein metrics.
The uniqueness part is known to have a positive answer by recent results of Biquard [17,Thm. 4] and Anderson-Herzlich [9]. The existence was settled in a paper by Koiso [42] inthe real analytic setting. As we were unaware of that paper, in a previous draft of thiswork we had proved in detail that the answer to the existence part of the above Cauchyproblem is positive in the analytic setting (Theorem 2.1). We review the proof in Section 2and show that the answer is negative, in general, in the smooth setting (Proposition 2.5).Let us also mention that DeTurck [26] analyzed in the Riemannian setting the somewhatrelated problem of finding a metric with prescribed nonsingular Ricci tensor.
Extension of generalized Killing spinors to parallel spinors.
Our main focus in thispaper is the extension problem for spinors. In order to introduce it, we must recall somebasic facts about restrictions of spin bundles to hypersurfaces. If Z is a Riemannian spinmanifold, any oriented hypersurface M ⊂ Z inherits a spin structure and it is well-knownthat the restriction to M of the complex spin bundle Σ Z if n is even (resp. Σ + Z if n isodd) is canonically isomorphic to the complex spin bundle Σ M (cf. [15]). If W denotesthe Weingarten tensor of M , the spin covariant derivatives ∇ Z on Σ Z and ∇ g on Σ M arerelated by ([15, Eq. (8.1)])( ∇ Z X Ψ) | M = ∇ gX (Ψ | M ) − W ( X ) · (Ψ | M ) , ∀ X ∈ T M, (1.1)for all spinors (resp. half-spinors for n odd) Ψ on Z . We thus see that if Ψ is a parallelspinor on Z , its restriction ψ to any hypersurface M is a generalized Killing spinor on M ,i.e. it satisfies the equation ∇ gX ψ = W ( X ) · ψ, ∀ X ∈ T M, (1.2)and the symmetric tensor W , called the stress-energy tensor of ψ , is just the Weingartentensor of the hypersurface M . It is natural to ask whether the converse holds: (Q2): If ψ is a generalized Killing spinor on M n , does there exist an iso-metric embedding of M into a spin manifold ( Z n +1 , g Z ) carrying a parallelspinor Ψ whose restriction to M is ψ ? This question is the Cauchy problem for metrics with parallel spinors asked in [15].The answer is known to be positive in several special cases: if the stress-energy tensor W of ψ is the identity [12], if W is parallel [46] and if W is a Codazzi tensor [15]. Even earlier,Friedrich [30] had worked out the 2-dimensional case n + 1 = 2 + 1, which is also coveredby [15, Thm. 8.1] since on surfaces the stress-energy of a generalized Killing spinor isautomatically a Codazzi tensor. Some related embedding results were also obtained byKim [41], Lawn–Roth [43] and Morel [47]. The common feature of each of these cases isthat one can actually construct in an explicit way the “ambient” metric g Z on the product( − ε, ε ) × M .Our aim is to show that the same is true more generally, under the sole additionalassumption that ( M, g ) and W are analytic. Theorem 1.1.
Let ψ be a spinor field on an analytic spin manifold ( M n , g ) , and W ananalytic field of symmetric endomorphisms of T M . Assume that ψ is a generalized Killing HE CAUCHY PROBLEMS FOR EINSTEIN METRICS AND PARALLEL SPINORS 3 spinor with respect to W , i.e. it satisfies (1.2) . Then there exists a unique metric g Z ofthe form g Z = dt + g t , with g = g , on a sufficiently small neighborhood Z of { } × M inside R × M such that ( Z , g Z ) , endowed with the spin structure induced from M , carriesa parallel spinor Ψ whose restriction to M is ψ . In particular, the solution g Z must be Ricci-flat. Einstein manifolds are analytic butof course hypersurfaces can lose this structure so our hypothesis is restrictive. Note thatEinstein metrics with smooth initial data can be constructed for small time as constantsectional curvature metrics when the second fundamental form is a Codazzi tensor, see [15,Thm. 8.1]. In particular in dimensions 1 + 1 and 2 + 1 Theorem 1.1 remains valid in thesmooth category since the tensor W associated to a generalized Killing spinor is automat-ically a Codazzi tensor in dimensions 1 and 2.The situation changes drastically in higher dimensions for smooth (instead of analytic)generalized Killing spinors. What we can still achieve then is to solve the Einstein equation(and the parallel spinor equation) in Taylor series near the initial hypersurface. Moreprecisely, starting from a smooth hypersurface ( M, g ) with prescribed Weingarten tensor W we prove that there exist formal Einstein metrics g Z such that W is the second fundamentalform at t = 0, i.e., we solve the Einstein equation modulo rapidly vanishing errors. Guidedby the analytic and the low dimensional ( n = 1 or n = 2) cases, one could be tempted toguess that actual germs of Einstein metrics do exist for any smooth initial data. Howeverthis turns out to be false. Counterexamples were found very recently in some particularcases in dimensions 3 and 7 by Bryant [20]. We give a general procedure to constructcounterexamples in all dimensions in Section 4.Note that several particular instances of Theorem 1.1 have been proved in recent years,based on the characterization of generalized Killing spinors in terms of exterior forms in lowdimensions. Indeed, in dimensions 5, 6 and 7, generalized Killing spinors are equivalent toso-called hypo , half-flat and co-calibrated G structures respectively. In [40] Hitchin provedthat the cases 6 + 1 and 7 + 1 can be solved up to the local existence of a certain gradientflow. Later on, Conti and Salamon [22], [23] treated the cases 5 + 1, 6 + 1 and 7 + 1 in theanalytical setting, cf. also [21] for further developments.A construction related to the Cauchy problem for Einstein metrics has been studiedstarting with the work of Fefferman-Graham [28] concerning asymptotically hyperbolicPoincar´e-Einstein metrics. The starting hypersurface ( M n , g ) is then at infinite distancefrom the manifold Z = (0 , ε ) × M , the metric g Z being conformal to a metric ¯ g of class C n − on the manifold with boundary Z = [0 , ε ) × M : g Z = x − ¯ g, ¯ g = dx + g x such that the conformal factor x is precisely the distance function to the boundary x = 0with respect to ¯ g . The metric is required to be Einstein of negative curvature up to an errorterm which vanishes with all derivatives at infinity. Such a metric always exists; when n isodd, it is smooth down to x = 0 and its Taylor series at infinity is determined by the initialmetric g and the symmetric transverse traceless tensor g n appearing as coefficient of x n BERND AMMANN, ANDREI MOROIANU, AND SERGIU MOROIANU in g x , while in even dimensions some logarithmic terms must be allowed, more precisely g x is smooth as a function of x and x n log x .Let us stress that existence results of Einstein metrics with prescribed first fundamentalform and Weingarten tensor clearly cannot hold globally in general (Example 2.7). Counterexamples in the smooth setting.
In the second part of the paper (Section 4)we apply the existence results from the analytic setting to prove nonexistence of solutionsfor certain smooth initial data in any dimension at least 3.The argument goes along the lines of works of the first author and his collaborators onthe Yamabe problem and the mass endomorphism. We consider the functional F ( φ ) := h D φ, φ i L k D φ k L n/ ( n +1) defined on the C spinor fields φ on a compact connected Riemannian spin manifold ( M, g )which are not in the kernel of the Dirac operator D . If the infimum of the lowest positiveeigenvalue of the Dirac operator in the volume-normalized conformal class of g is strictlylower than the corresponding eigenvalue for the standard sphere (Condition (4.5) below),this functional attains its supremum in a spinor ψ of regularity C ,α . Moreover, ψ issmooth outside its zero set.To construct g satisfying Condition (4.5) we fix p ∈ M and we look at metrics on M which are flat near p . If the topological index of M vanishes in KO − n ( pt ), then forgeneric such metrics the associated Dirac operator is invertible. The mass endomorphismat p is defined as the constant term in the asymptotic expansion of the Green kernel of D near p . Again for generic metrics, this mass endomorphism is non-zero, which by a resultof [7] ensures the technical Condition (4.5) for generic metrics which are flat near p . Byconstruction this class of metrics contains metrics which are not conformally flat on someopen subset of M , i.e., whose Schouten tensor (in dimension 3), resp. Weyl curvature (inhigher dimensions) is nonzero on some open set. We assume g was chosen with theseproperties.We return now to the spinor ψ maximizing the functional F . The Euler-Lagrangeequation of F at ψ can be reinterpreted as follows: the Dirac operator with respect to theconformal metric g := | ψ | / ( n − g admits an eigenspinor of constant length 1, ψ := ψ | ψ | .If the dimension n equals 3, by algebraic reasons this spinor field must be a generalizedKilling spinor with stress-energy tensor W of constant trace.The metric g is defined on the complement M ∗ of the zero set of ψ . This set is open,connected and dense in M (Lemmata 4.6 and 4.9). Recall that g was chosen such thatits Schouten tensor vanishes identically on an open set of M and is nonzero on anotheropen set. Then the same remains true on M ∗ , and therefore on M ∗ there exists no analyticmetric in the conformal class of g . In particular, the metric g = | ψ | / ( n − g cannot beanalytic. HE CAUCHY PROBLEMS FOR EINSTEIN METRICS AND PARALLEL SPINORS 5
Assuming now that Theorem 2.1 continues to hold for smooth initial data, we couldapply it to ( M ∗ , g, W ) to get an embedding in a Ricci-flat (hence analytic) Riemannianmanifold ( Z , g Z ), with second fundamental form W . Since the trace of W is constant byconstruction, M would have constant mean curvature, which would imply that it wereanalytic (Lemma 4.16), contradicting the non-analyticity proved above.The above construction actually yields counterexamples to the Cauchy problem for Ricci-flat metrics in the smooth setting in any dimension n ≥
3, by taking products with flatspaces, see Lemma 4.28.
Acknowledgements.
It is a pleasure to thank Olivier Biquard, Gilles Carron, MattiasDahl, Paul Gauduchon, Colin Guillarmou, Christophe Margerin, Yann Rollin and Jean-Marc Schlenker for helpful discussions. We thank the DFG-Graduiertenkolleg GRK 1692Regensburg for its support. AM was partially supported by the contract ANR-10-BLAN0105 “Aspects Conformes de la G´eom´etrie” and by the LEA “MathMode”. SM was par-tially supported by the contract PN-II-RU-TE-2011-3-0053 and by the LEA “MathMode”.He thanks the CMLS at the Ecole Polytechnique for its hospitality during the writing ofthis paper. 2.
The Cauchy problem for Einstein metrics
Let ( Z , g Z ) be an oriented Riemannian manifold of dimension n + 1, and M an ori-ented hypersurface with induced Riemannian metric g := g Z | M . We start by fixing somenotations. Denote by ∇ Z and ∇ g the Levi-Civita covariant derivatives on ( Z , g Z ) and( M, g ), by ν the unit normal vector field along M compatible with the orientations, andby W ∈ End(
T M ) the Weingarten tensor defined by ∇ Z X ν = − W ( X ) , ∀ X ∈ T M. (2.1)Using the normal geodesics issued from M , the metric on Z can be expressed in a neigh-borhood Z of M as g Z = dt + g t , where t is the distance function to M and g t is a familyof Riemannian metrics on M with g = g (cf. [15]). The vector field ν extends to Z as ν = ∂/∂t and (2.1) defines a symmetric endomorphism on Z which can be viewed as afamily W t of endomorphisms of M , symmetric with respect to g t , and satisfying (cf. [15,Equation (4.1)]): g t ( W t ( X ) , Y ) = − ˙ g t ( X, Y ) , ∀ X, Y ∈ T M. (2.2)By [15, Equations (4.5)–(4.8)], the Ricci tensor and the scalar curvature of Z satisfy forevery vectors X, Y ∈ T M
Ric Z ( ν, ν ) = tr( W t ) − tr g t (¨ g t ) , (2.3) Ric Z ( ν, X ) = d tr( W t )( X ) + δ g t ( W )( X ) , (2.4) Ric Z ( X, Y ) = Ric g t ( X, Y ) + 2 g t ( W t X, W t Y ) + tr( W t ) ˙ g t ( X, Y ) − ¨ g t ( X, Y ) , (2.5) Scal Z = Scal g t + 3tr( W t ) − tr ( W t ) − tr g t (¨ g t ) . (2.6) BERND AMMANN, ANDREI MOROIANU, AND SERGIU MOROIANU where in (2.4) the divergence operator δ g : End( T M ) → T ∗ M is defined in a local g -orthonormal basis { e i } of T M by(2.7) δ g ( A )( X ) = − n X i =1 g (( ∇ ge i A )( e i ) , X ) . Using (2.3) and (2.6) we get(2.8) − Z ( ν, ν ) + Scal Z = Scal g t + tr( W t ) − tr ( W t ) . Assume now that the metric g Z is Einstein with scalar curvature ( n +1) λ , i.e. Ric Z = λg Z .Evaluating (2.4) and (2.8) at t = 0 yields d tr( W ) + δ g W = 0 , (2.9) Scal g + tr( W ) − tr ( W ) = ( n − λ. (2.10)If g t : End( T M ) → T ∗ M ⊗ T ∗ M is the isomorphism defined by g t ( A )( X, Y ) := g t ( A ( X ) , Y )and g − t : T ∗ M ⊗ T ∗ M → End(
T M ) denotes its inverse, then taking (2.3) into account,(2.5) reads(2.11) ¨ g t = 2Ric g t + ˙ g t ( g − t ( ˙ g t ) · , · ) − tr( g − t ( ˙ g t )) ˙ g t − λg t . Using the Cauchy-Kowalewskaya theorem, Koiso proved the following existence andunique continuation result for Einstein metrics starting from an analytic metric and ananalytic stress-energy tensor satisfying the above constraints.
Theorem 2.1 ([42]) . Let ( M n , g ) be an analytic Riemannian manifold and let W be ananalytic symmetric endomorphism field on M satisfying (2.9) and (2.10) . Then for ε > ,there exists a unique analytic germ near { } × M of an Einstein metric g Z with scalarcurvature ( n + 1) λ of the form g Z = dt + g t on Z := R × M , with g = g , whoseWeingarten tensor at t = 0 is W .Sketch of proof. In equation (2.11) the only term involving partial derivatives of the metric g t along M is Ric g t , which is an analytic expression in g t and its first and second orderderivatives along M which does not involve any derivative with respect to t .The second order Cauchy-Kowalewskaya theorem (see e.g. [24]) shows that for every x ∈ M there exists a neighborhood V x ∋ x and some ε x > g = g, ˙ g = − W has a unique analytic solution g t on ( − ε x , ε x ) × V x . Using the uniqueness of solutionsfor systems of linear ODE’s, one can then prove that g Z = dt + g t is Einstein withscalar curvature ( n + 1) λ . By uniqueness, these metrics patch up to a global metric near M × { } . (cid:3) As direct consequences of Theorem 2.1, one obtains the following embedding results foranalytic metrics and conformal structures:
HE CAUCHY PROBLEMS FOR EINSTEIN METRICS AND PARALLEL SPINORS 7
Corollary 2.2.
Let ( M n , g ) be an analytic Riemannian manifold of constant scalar cur-vature. Then for every λ ≤ Scal g n − , M can be isometrically embedded as a totally umbilicalhypersurface in an Einstein manifold ( Z n +1 , g Z ) with Ricci constant λ , i.e., Ric Z = λg Z .Proof. The tensor W := α id satisfies Equations (2.9), (2.10) for α = q Scal g n ( n − − λn (cid:3) A conformal structure c on a manifold M is called analytic if there exists an analyticatlas on M and an analytic metric g in c (see Definition 4.12 below). Corollary 2.3.
Let ( M n , c ) be a compact analytic conformal manifold of Yamabe invariant Y ( M, c ) . Then M can be conformally embedded as a totally umbilical hypersurface in anEinstein manifold ( Z n +1 , g Z ) with Ric Z = λg Z for every λ with sign( λ ) ≤ sign( Y ( M, c )) .Proof. Let g be some analytic metric in c . Using the solution to the Yamabe problem forcompact manifolds we get a unit volume metric g = u / ( n − g ∈ c with constant scalarcurvature Scal g = Y ( M, c ). The function u satisfies a linear elliptic second order differentialequation (the conformal Laplacian) with analytic coefficients, so g is analytic. The resultnow follows from the previous corollary, after a suitable constant rescaling of g . (cid:3) The Cauchy problem for smooth initial data.
It was proven recently by Biquard[17, Thm. 4] and Anderson-Herzlich [9] that even in the C ∞ setting, given a hypersurface M ⊂ Z , a Riemannian metric on M and a field of symmetric endomorphisms W , thereexists (up to diffeomorphisms preserving the hypersurface) at most one Einstein metricon Z with Weingarten tensor W along M .The small-time existence however is known to fail in general for elliptic (even linear)Cauchy problems with C ∞ initial data. In the particular case of the Cauchy problemfor Einstein metrics, we first remark that in small dimensions the short-time existence isalways guaranteed by the construction of an explicit solution in the smooth (and actuallyeven C ) setting.Indeed, in dimension 1 we can embed any curve ( M, g ) in a constant curvature surfacewith prescribed extrinsic curvature function (identified with the scalar Weingarten tensor) W . In this case, the constraint equations are empty, and the metric is explicitly given by[15, Theorem 7.2].Similarly, in dimension n = 2, the C initial value problem can always be solved forsmall time: Proposition 2.4.
Let M be a surface with C Riemannian metric g , and let W be a C symmetric field of endomorphisms on M satisfying (2.9) and (2.10) for some λ ∈ R .Then there exists a metric g Z of constant sectional curvature κ = λ/ on a neighborhoodof { } × M inside Z := R × M of the form g Z = dt + g t , with g = g , whose Weingartentensor at t = 0 is W .Proof. Direct application of [15, Theorem 7.2]. Namely, in dimension 2 the hypotheses(2.9), (2.10) are equivalent to [15, Eq. (7.3)] resp. [15, Eq. (7.4)] with κ = λ/
2. It follows,
BERND AMMANN, ANDREI MOROIANU, AND SERGIU MOROIANU at least in the smooth case, that g t can be constructed explicitly in terms of g and W such that g Z has constant sectional curvature κ . It remains to note that the proof of [15,Theorem 7.2] remains valid when g and W are of class C . (cid:3) In higher dimensions n ≥ Proposition 2.5.
A Riemannian manifold ( M n , g ) of constant scalar curvature can beisometrically embedded in an Einstein manifold ( Z n +1 , g Z ) with Weingarten tensor W = α id along M if and only if g is analytic.Proof. The tensor W satisfies Equations (2.9), (2.10) for Scal g = ( n − λ + nα ). The “if”part thus follows from Theorem 2.1. Conversely, if such an embedding exists, then ( M, g )is a constant mean curvature hypersurface in ( Z , g Z ), so g has to be analytic by Lemma4.16 below. (cid:3) Note that a metric with constant scalar curvature is automatically analytic in dimen-sions 1 and 2. Examples of non-analytic constant scalar curvature metrics in dimensionsat least 3 can be easily constructed: perturb the round metric on S n to a metric g whichis non-conformally flat on some open set and conformally flat on some other open set andchoose a constant scalar curvature metric in the conformal class of g using the solution ofthe Yamabe problem. Formal solution in the smooth case.
The previous arguments show that without thehypothesis that g and W are analytic, the nonlinear PDE system (2.11) has no solutionin general. However, it is rather evident from (2.11) that the full Taylor series of g Z is recursively determined by its first two coefficients, which are g and W . Let ˙ C ∞ ( Z )denote the space of tensors vanishing at M together with all their derivatives. By theBorel lemma (see e.g. [31]), there exists a metric g Z such that its Ricci tensor satisfies theEinstein equation in the tangential directions modulo ˙ C ∞ ( Z ). Then we can easily showrecursively that the right-hand sides of Equations (2.4) and (2.8) vanish modulo ˙ C ∞ ( Z ).Thus g Z is Einstein modulo ˙ C ∞ ( Z ). Proposition 2.6.
Let ( M n , g ) be a smooth Riemannian manifold and let W be a smoothsymmetric field of endomorphisms of T M satisfying (2.9) and (2.10) . Then there existson Z := ( − ε, ε ) × M a metric g Z of the form g Z = dt + g t , with g = g , whose Weingartentensor at t = 0 is W , and such that Ric Z − λg Z ∈ ˙ C ∞ ( Z ) . Moreover, g Z is unique up to ˙ C ∞ ( Z ) . A counterexample to long-time existence.
The preceding case of dimension 2 + 1hints that in general the Einstein metric g Z cannot be extended on a complete manifoldcontaining M as a hypersurface. This sort of question is rather different from the argumentsof this paper so we will only give an counterexample in dimension 1 + 1 where global HE CAUCHY PROBLEMS FOR EINSTEIN METRICS AND PARALLEL SPINORS 9 existence for the solution to the Cauchy problem fails. We restrict ourselves to the case ofRicci-flat metrics, which means vanishing Gaussian curvature in this dimension.
Example . Let Z be the incomplete flat surface obtained from C ∗ (or from the comple-ment of a small disk in C ) by the following cut-and-paste procedure: cut along the positivereal axis, then glue again after a translation of length l >
0. More precisely, x + is identifiedwith ( x + l ) − for all x > ε . The resulting surface Z is clearly smooth and has a smoothflat metric including along the gluing locus. The unit circle in R gives rise to a curvein Z of curvature 1 and length 2 π with different endpoints 1 − and (1 + l ) − . In a completeflat surface, a curve of curvature 1 and length 2 π must be closed (in fact smooth, since itslift to the universal cover must be a circle). Therefore, the surface Z cannot be embeddedin any complete flat surface. In particular, for any closed curve in Z circling around thesingular locus, the interior cannot be continued to a compact flat surface with boundary.3. Spinors on Ricci-flat manifolds
We come now to parallel and generalized Killing spinors, our main object of interestin this paper. We keep the notation from the previous section. Our starting point is thefollowing corollary of Theorem 2.1:
Corollary 3.1.
Assume that ( M n , g ) is an analytic spin manifold carrying a non-trivialgeneralized Killing spinor ψ with analytic stress-energy tensor W . Then in a neighborhoodof { }× M in Z := R × M there exists a unique Ricci-flat metric g Z of the form g Z = dt + g t whose Weingarten tensor at t = 0 is W .Proof. We just need to check that the constraints (2.9), (2.10) are a consequence of (1.2).In order to simplify the computations, we will drop the reference to the metric g anddenote respectively by ∇ , R , Ric and Scal the Levi-Civita covariant derivative, curvaturetensor, Ricci tensor and scalar curvature of ( M, g ). As usual, { e i } will denote a local g -orthonormal basis of T M .We will use the following two classical formulas in Clifford calculus. The first one is thefact that the Clifford contraction of a symmetric tensor A only depends on its trace:(3.1) n X i =1 e i · A ( e i ) = − tr( A ) . The second formula expresses the Clifford contraction of the spin curvature in terms of theRicci tensor ([11], p. 16): n X i =1 e i · R X,e i ψ = − Ric( X ) · ψ, ∀ X ∈ T M, ∀ ψ ∈ Σ M. (3.2)Let now ψ be a non-trivial generalized Killing spinor satisfying (1.2). Being parallel withrespect to a modified connection on Σ M , ψ is nowhere vanishing (and actually of constantnorm). Taking a further covariant derivative in (1.2) and skew-symmetrizing yields R X,Y ψ = ( W ( Y ) · W ( X ) − W ( X ) · W ( Y )) · ψ + (( ∇ X W )( Y ) − ( ∇ Y W )( X )) · ψ for all X, Y ∈ T M . In this formula we set Y = e i , take the Clifford product with e i andsum over i . From (3.1) and (3.2) we getRic( X ) · ψ = − n X i =1 e i · ( W ( e i ) · W ( X ) − W ( X ) · W ( e i )) · ψ − n X i =1 e i · (( ∇ X W )( e i ) − ( ∇ e i W )( X )) · ψ = tr( W ) W ( X ) · ψ + n X i =1 (cid:0) − W ( X ) · e i − g ( W ( X ) , e i ) (cid:1) · W ( e i ) · ψ + ∇ X (tr( W )) ψ + n X i =1 e i · ( ∇ e i W )( X ) · ψ. whence(3.3) Ric( X ) · ψ = tr( W ) W ( X ) · ψ − W ( X ) · ψ + X (tr( W )) ψ + n X i =1 e i · ( ∇ e i W )( X ) · ψ. We set X = e j in (3.3), take the Clifford product with e j and sum over j . Using (3.1)again we obtain − Scal ψ = − tr ( W ) ψ + tr( W ) ψ + ∇ (tr( W )) · ψ + n X i,j =1 e j · e i · ( ∇ e i W )( e j ) · ψ = − tr ( W ) ψ + tr( W ) ψ + d tr( W ) · ψ + n X i,j =1 ( − e i · e j − δ ij ) · ( ∇ e i W )( e j ) · ψ = − tr ( W ) ψ + tr( W ) ψ + 2 d tr( W ) · ψ + 2 δW · ψ, which implies simultaneously (2.9) and (2.10) (indeed, if f ψ = X · ψ for some real f andvector X , then −| X | ψ = X · X · ψ = X · ( f ψ ) = f ψ , so both f and X vanish). (cid:3) Theorem 3.2.
Let ( Z , g Z ) be a Ricci-flat spin manifold with Levi-Civita connection ∇ Z and let M ⊂ Z be any oriented analytic hypersurface. Assume there exists some spinor ψ ∈ C ∞ (Σ Z | M ) which is parallel along M : ∇ Z X ψ = 0 , ∀ X ∈ T M ⊂ T Z . (3.4) Assume moreover that the application π ( M ) → π ( Z ) induced by the inclusion is onto.Then there exists a parallel spinor Ψ ∈ C ∞ (Σ Z ) such that Ψ | M = ψ .Proof. Any Ricci-flat manifold is analytic, cf. [27], [16], thus the analyticity of M makessense. The proof is split in two parts. HE CAUCHY PROBLEMS FOR EINSTEIN METRICS AND PARALLEL SPINORS 11
Local extension.
Let ν denote the unit normal vector field along M . Every x ∈ M hasan open neighborhood V in M such that the exponential map ( − ε, ε ) × V → Z , ( t, y ) exp y ( tν ) is well-defined for some ε >
0. Its differential at (0 , x ) being the identity, one canassume, by shrinking V and choosing a smaller ε if necessary, that it maps ( − ε, ε ) × V diffeomorphically onto some open neighborhood U of x in Z . We extend the spinor ψ to aspinor Ψ on U by parallel transport along the normal geodesics exp y ( tν ) for every fixed y .It remains to prove that Ψ is parallel on U in horizontal directions.Let { e i } be a local orthonormal basis along M . We extend it on U by parallel transportalong the normal geodesics, and notice that { e i , ν } is a local orthonormal basis on U . Moregenerally, every vector field X along V gives rise to a unique horizontal vector field, alsodenoted X , on U satisfying ∇ ν X = 0. For every such vector field we get(3.5) ∇ Z ν ( ∇ Z X Ψ) = R Z ( ν, X )Ψ + ∇ Z [ ν,X ] Ψ = R Z ( ν, X )Ψ + ∇ Z W ( X ) Ψ . Since Z is Ricci-flat, (3.2) applied to the local orthonormal basis { e i , ν } of Z yields(3.6) 0 = Ric Z ( X ) · Ψ = n X i =1 e i · R Z ( e i , X )Ψ + ν · R Z ( ν, X )Ψ . We take the Clifford product with ν in this relation, differentiate again with respect to ν and use the second Bianchi identity to obtain: ∇ Z ν ( R Z ( ν, X )Ψ) = ∇ Z ν ν · n X i =1 e i · R Z ( e i , X )Ψ ! = ν · n X i =1 e i · ( ∇ Z ν R Z )( e i , X )Ψ= ν · n X i =1 e i · (cid:0) ( ∇ Z e i R Z )( ν, X )Ψ + ( ∇ Z X R Z )( e i , ν )Ψ (cid:1) , whence ∇ Z ν ( R Z ( ν, X )Ψ) = ν · n X i =1 e i · (cid:0) ∇ Z e i ( R Z ( ν, X )Ψ) + R Z ( W ( e i ) , X )Ψ − R Z ( ν, ∇ Z e i X )Ψ − R Z ( ν, X ) ∇ Z e i Ψ + ∇ Z X ( R Z ( e i , ν )Ψ) − R Z ( ∇ Z X e i , ν )Ψ+ R Z ( e i , W ( X ))Ψ − R Z ( e i , ν ) ∇ Z X Ψ (cid:1) . (3.7)Let ν ⊥ denote the distribution orthogonal to ν on U and consider the sections A, B ∈ C ∞ (( ν ⊥ ) ∗ ⊗ Σ U ) and C ∈ C ∞ (Λ ( ν ⊥ ) ∗ ⊗ Σ U ) defined for all X, Y ∈ ν ⊥ by A ( X ) := ∇ Z X Ψ , B ( X ) := R Z ( ν, X )Ψ , C ( X, Y ) := R Z ( X, Y )Ψ . We have noted that the metric g Z is analytic since it is Ricci-flat. From the assumptionthat M is analytic and that ψ is parallel along M it follows that Ψ, and thus the tensors A , B and C , are analytic.Equations (3.5) and (3.7) read in our new notation:(3.8) ( ∇ Z ν A )( X ) = B ( X ) + A ( W ( X )) , and ( ∇ Z ν B )( X ) = ν · n X i =1 e i · (cid:0) ( ∇ Z e i B )( X ) + C ( W ( e i ) , X ) − R Z ( ν, X ) A ( e i ) − ( ∇ Z X B )( e i ) + C ( e i , W ( X )) − R Z ( e i , ν ) A ( X ) (cid:1) . (3.9)Moreover, the second Bianchi identity yields( ∇ Z ν C )( X, Y ) =( ∇ Z ν R Z )( X, Y )Ψ = ( ∇ Z X R Z )( ν, Y )Ψ + ( ∇ Z Y R Z )( X, ν )Ψ= ∇ Z X ( R Z ( ν, Y )Ψ) − R Z ( ∇ Z X ν, Y )Ψ − R Z ( ν, ∇ Z X Y )Ψ − R Z ( ν, Y ) ∇ Z X Ψ − ∇ Z Y ( R Z ( ν, X )Ψ) + R Z ( ∇ Z Y ν, X )Ψ + R Z ( ν, ∇ Z Y X )Ψ + R Z ( ν, X ) ∇ Z Y Ψ=( ∇ Z X B )( Y ) + C ( W ( X ) , Y ) − R Z ( ν, Y ) ∇ Z X Ψ − ( ∇ Z Y B )( X ) + C ( X, W ( Y )) + R Z ( ν, X ) ∇ Z Y Ψ , thus showing that( ∇ Z ν C )( X, Y ) =( ∇ Z X B )( Y ) + C ( W ( X ) , Y ) − R Z ( ν, Y )( A ( X )) − ( ∇ Z Y B )( X ) + C ( X, W ( Y )) + R Z ( ν, X )( A ( Y )) . (3.10)The hypothesis (3.4) is equivalent to A = 0 for t = 0. Differentiating this again in thedirection of M and skew-symmetrizing yields C = 0 for t = 0. Finally, (3.6) shows that B = 0 for t = 0. We thus see that the section S := ( A, B, C ) vanishes on along thehypersurface { } × V of U .The system (3.9)–(3.10) is a linear PDE for S and the hypersurfaces t = constantare clearly non-characteristic. The Cauchy-Kowalewskaya theorem shows that S vanisheseverywhere on U . In particular, A = 0 on U , thus proving our claim. Global extension.
Now we prove that there exists a parallel spinor Ψ ∈ C ∞ (Σ Z ) such thatΨ | M = ψ . Take any x ∈ M and an open neighborhood U like in Theorem 3.2 on which aparallel spinor Ψ extending ψ is defined. The spin holonomy group g Hol(
U, x ) thus preservesΨ x . Since any Ricci-flat metric is analytic (cf. [16, p. 145]), the restricted spin holonomygroup g Hol ( Z , x ) is equal to g Hol ( U, x ) for every x ∈ Z and for every open neighborhood U of x . By the local extension result proved above, g Hol ( U, x ) acts trivially on Ψ x , thusshowing that Ψ x can be extended (by parallel transport along every curve in ˜ Z startingfrom x ) to a parallel spinor ˜Ψ on the universal cover ˜ Z of Z . The deck transformationgroup acts trivially on ˜Ψ since every element in π ( Z , x ) can be represented by a curve in M (here we use the surjectivity hypothesis) and Ψ was assumed to be parallel along M .Thus ˜Ψ descends to Z as a parallel spinor. (cid:3) This result, together with Corollary 3.1 yields the solution to the analytic Cauchy prob-lem for parallel spinors stated in Theorem 1.1.
HE CAUCHY PROBLEMS FOR EINSTEIN METRICS AND PARALLEL SPINORS 13 Construction of generalized Killing spinors
The goal of this section is to describe a method which yields generalized Killing spinorson many 3-dimensional spin Riemannian manifolds. We will obtain both analytic and non-analytic generalized Killing spinors. The analytic ones will yield examples for applyingTheorem 1.1. The non-analytic ones only yield a formal Taylor series in the sense ofProposition 2.6, and we will show that in general this solution is not the Taylor series ofa Ricci-flat metric. Thus we see that the analyticity assumption in Theorem 1.1 cannotbe removed. The method consists in combining techniques developed elsewhere. We statebelow the relevant results and briefly explain the underlying ideas.Note that further examples of manifolds with generalized Killing spinors which can notbe embedded as hypersurfaces in manifolds with parallel spinors were recently constructed(although not explicitly stated), by Bryant [20] in the context of K -structures satisfyingthe so-called weaker torsion condition .4.1. Minimizing the first Dirac eigenvalue in a conformal class.
In [2] and [3] thefollowing problem was studied: Suppose M is an n -dimensional compact spin manifold, n ≥ g on M let D g be the Diracoperator on M . The spectrum of D g is discrete, and all eigenvalues have finite multiplicity.The first positive eigenvalue of D g will be denoted by λ +1 ( g ). In general, the dimensionof the kernel of D g depends on g , and on many manifolds (in particular on all compactspin manifolds of dimension n ≡ , , , n ≥
3) metrics g i are known such that g i → g in the C ∞ -topology, dim ker D g i < dim ker D g and λ +1 ( g i ) →
0. Thus g λ +1 ( g ) isnot continuous when defined on the set of all metrics.We now fix a conformal class [ g ] on M , and only consider metrics g ∈ [ g ]. Then theabove properties change essentially. Due to the conformal behavior of the Dirac opera-tor, the dimension of the kernel of D g is constant on [ g ], and furthermore [ g ] → R + , g λ +1 ( g ) is continuous in the C -topology. For any positive real number α one has λ +1 ( α g ) = α − λ +1 ( g ). The normalized first positive eigenvalue function [ g ] → (0 , ∞ ), g λ +1 ( g )vol( M, g ) /n , is thus scaling invariant and continuous in the C -topology. It isunbounded from above, see [8], and bounded from below by a positive constant, see [45]in the case ker D g = 0 and [1, 3] for the general case. We introduce(4.1) λ +min ( M, [ g ]) := inf g ∈ [ g ] λ +1 ( g )vol( M, g ) /n > . If there is a metric of positive scalar curvature in [ g ], then the Yamabe constant(4.2) Y ( M, [ g ]) := inf g ∈ [ g ] vol( M, g ) (2 − n ) /n Z M Scal g dv g is positive, and Hijazi’s inequality [36, 37] then yields(4.3) λ +min ( M, [ g ]) ≥ n n − Y ( M, [ g ]) . Example . If (
M, g ) = S n is the sphere S n with the conformal structure given bythe standard metric σ n of volume ω n , then Obata’s theorem [50, Prop. 6.2] implies that the infimum in (4.2) is attained in g = σ , and thus Y ( S n ) = n ( n − ω /nn . We obtain λ +min ( S n ) ≥ n ω /nn . On the other hand ( M, σ ) carries a Killing spinor to the Killing constant − /
2, thus λ +1 ( σ ) = n . As a consequence, equality is attained in (4.3), the infimum in (4.1)is attained in g = σ and λ +min ( S n ) = n ω /nn .Now let ( M, g ) be again arbitrary. By “blowing up a sphere” one can show that λ +min ( M, [ g ]) ≤ λ +min ( S n ), see [1, 6]. This inequality should be seen as a spinorial analogueof Aubin’s inequality between the Yamabe constants Y ( M, [ g ]) ≤ Y ( S n ) = n ( n − ω /nn .For the Yamabe constants one even gets a stronger statement: If ( M, g ) is not conformalto the round sphere, then(4.4) Y ( M, [ g ]) < Y ( S n ) . This inequality leads to a solution of the Yamabe problem, see [44]. It was proved in somecases by Aubin [10]. Later Schoen and Yau [52, 53] could solve the remaining cases, usingthe positive mass theorem.It is thus natural to ask the following question which is still open in general.
Question 4.2.
Under the assumption that ( M, [ g ]) is not conformal to ( S n ) , n ≥ , doesthe inequality (4.5) λ +min ( M, [ g ]) < λ +min ( S n ) . always hold? We will explain below that many Riemannian manifolds, in particular “generic” metricson compact spin 3-dimensional manifolds, do satisfy (4.5). It is interesting to notice thatusing (4.3) the inequality (4.5) would imply (4.4) without referring to the positive masstheorem.In analogy to the Yamabe problem which consists in finding a smooth metric attainingthe infimum in (4.2), one can try to find a metric attaining the infimum in (4.1). If thisinfimum is achieved in a metric g ∈ [ g ], then the corresponding Euler-Lagrange equationprovides the existence of an eigenspinor ψ of constant length of eigenvalue λ +1 ( g ). Indimension n = 3, such constant-length eigenspinors are generalized Killing spinors, seeSubsection 4.3, and – as said above – it is the goal of this section to construct generalizedKilling spinors.Unfortunately, it is unclear whether the infimum in (4.1) can be achieved by a (smooth)metric. However, if we assume that (4.5) holds, and if we allow degenerations in the confor-mal factor, the infimum is attained. To explain the nature of these possible degenerationsprecisely, we introduce the following. A generalized metric in the conformal class [ g ] is ametric of the form f g where f is continuous on M and smooth on M ∗ := M \ f − (0).Moreover, we only admit such generalized metrics for which M ∗ is dense in M . The set ofall such admissible generalized metric associated to the conformal class [ g ] will be denotedby [ g ]. HE CAUCHY PROBLEMS FOR EINSTEIN METRICS AND PARALLEL SPINORS 15
Remark . The above definitions are slight more restrictive than in [2], but sufficient forthe purpose of the present article and didactically simpler. For example, the conditionthat M ∗ is supposed to be dense, guarantees that [ g ] ∩ [ g ] = ∅ if g and g are notconformal.The functions λ , vol : [ g ] → R + extend continuously to functions [ g ] → R + , and theinfimum in (4.1) does not change when we replace [ g ] by [ g ]. We then have Theorem 4.4 ([2, Theorem 1.1(B)]) . Let ( M, g ) be a compact Riemannian spin manifoldof dimension n ≥ . There exists a generalized metric g ∈ [ g ] at which the infimum in (4.1)is attained. On ( M ∗ , g ) there exists a spinor ψ of constant length with Dψ = λ ( g ) ψ . The key idea in the proof of this theorem is to reformulate the problem of minimiz-ing (4.1) as a variational problem. For this we define F q ( φ ) = R h D g φ, φ i g dvol g k D g φ k L q ( g ) , µ g q := sup F g q ( φ ) , (4.6)where the supremum runs over all spinors φ of regularity C which are not in the kernelof D g . It was shown in [2, Prop. 2.3] that for q = nn +1 we have µ g n/ ( n +1) = λ +min ( M, [ g ]) . Furthermore the infimum in (4.1) is attained in a smooth metric g ∈ [ g ] if and only if thereis a nowhere vanishing spinor ψ which attains the supremum in (4.6). If the infimum isattained in g and the supremum in ψ , then both are related via(4.7) g = | D g ψ | / ( n +1) g . Proposition 4.5 ([2, Theorem 1.1 (A)]) . Under the condition (4.5) the supremum isattained in a spinor ψ of regularity C ,α for small α > . The strategy of proof is similar to the classical approach to the Yamabe problem ase.g. in [44]. A maximizing sequence for the functional will in general not converge, due toconformal invariance. One then defines “perturbed” or “regularized” modifications of thisfunctional such that their maximizing sequences converge to a maximizer. In a final stepone shows, assuming (4.5), that the maximizers of the perturbed functionals converge toa maximizer of the unperturbed functional.Let us now continue with the sketch of proof of Theorem 4.4. From Prop. 4.5 we knowthat the supremum of F is attained at some spinor ψ which satisfies an Euler-Lagrangeequation. By suitably rescaling ψ and by possibly adding an element of ker D g to ψ , theEuler-Lagrange equation reads D g ψ = λ +min ( M, [ g ]) | ψ | / ( n − ψ , k ψ k L n/ ( n − ( g ) = 1 . However, it is unclear whether D g ψ (or equivalently ψ ) has zeros or not, and thereforeif the metric g defined in (4.7) makes sense. We will show in the following subsection that the zero set is nowhere dense, in otherwords its complement is dense. Then g := | D g ψ | / ( n +1) g defines a generalized metric,and by naturally extending the definition of λ +1 to generalized metrics, we see that theinfimum in (4.1) is then attained in this generalized metric.Consistently with the above we set M ∗ := M \ ψ − (0). From the standard formulafor the behavior of the Dirac operator under conformal change (see e.g. [38]) the spinor ψ := ψ | ψ | on M ∗ satisfies D g ψ = λ +min ( M, [ g ]) ψ, | ψ | ≡ . This finishes the proof for Theorem 4.4, up to the density of M ∗ explained below.4.2. The zero set of the maximizing spinor.
The goal of this subsection is to studythe zero set of the maximizing spinor ψ from the previous section. Lemma 4.6.
Let ( M, g ) be a connected Riemannian spin manifold. Assume that a spinor φ of regularity C satisfies (4.8) D g φ = c | φ | r φ where r ≥ and c ∈ R . If φ vanishes on a non-empty open set, then it vanishes on M . Applying the lemma to φ := ψ r := 2 / ( n −
1) one obtains the density of M ∗ in M . Proof.
The lemma is a special case of the weak unique continuation principle [18]. Moreexactly we apply [18, Theorem 2.7] with D/ A = D g and P A ( φ, x ) := −| φ ( x ) | r . As φ is locally bounded, we see that x P A ( φ, x ) is locally bounded as well. Thus P A isan admissible perturbation in the sense of [18], and [18, Theorem 2.7] then yields theweak unique continuation principle for this equation which is exactly the statement of thelemma. (cid:3) We propose two conjectures around the above lemma.The first conjecture relies on the following remark: if r is an even integer, then | φ | r φ is a smooth function of φ , so the Main Theorem in [14] shows that the zero set of φ isa countably ( n − n −
2. Incontrast, if r is not an even integer, then B¨ar’s method of proof does not apply, but theresult seems likely to remain true. Conjecture . The zero set of any solution of (4.8) is of Hausdorff dimension at most n − r = 0, do not vanish anywhere; in other words they are everywhere non-zero.We conjecture that the same fact is true for r := n − . This constant r is special, asthen (4.8) and thus the zero set of φ is conformally invariant. HE CAUCHY PROBLEMS FOR EINSTEIN METRICS AND PARALLEL SPINORS 17
Conjecture . Let r := n − , and let M be connected. For generic conformal classes on M , any solution of (4.8) with φ M is connected, then the manifold M \ φ − (0) is connected.Fortunately, for the maximizing spinor ψ the following fact can be proven independentlyof the above conjectures: Lemma 4.9.
Assume M to be connected. Let ψ be the maximizing spinor provided byProposition 4.5. Then M ∗ = M \ ψ − (0) is connected.Proof. Assume that there exists a partition M ∗ = Ω ⊔ Ω into non-empty disjoint opensets. We define the continuous spinor ψ by ψ | Ω := ψ | Ω and ψ | M \ Ω : ≡
0. Then k ψ k L n/ ( n − < k ψ k L n/ ( n − . As a first step we prove by contradiction that ψ is C , orequivalently that ∇ ψ = 0 on ∂ Ω .Suppose that there existed x ∈ ∂ Ω ∩ ∂ Ω such that ∇ ψ is non-zero in x . Becauseof ( Dφ )( x ) = 0 the map T x M → Σ x M , X
7→ ∇ X ψ has rank at least 2. The implicitfunction theorem then implies that there is a connected open neighborhood U of x anda submanifold S ⊂ U of codimension 2 such that ψ − (0) ∩ U ⊂ S . This implies that U \ S ⊂ Ω . One easily concludes that S ∩ Ω = ∅ , thus we obtain the contradiction x ∂ Ω .We have proven that ψ is C , and thus ψ is a solution to D g ψ = λ +min ( M, [ g ]) | ψ | / ( n − ψ , < k ψ k L n/ ( n − ( g ) < . A straightforward calculation then yields F n/ ( n +1) ( ψ ) > λ +min ( M, [ g ]) = µ g n/ ( n +1) which contradicts the definition of µ g n/ ( n +1) . (cid:3) From eigenspinors of constant length to generalized Killing spinors.
In thissection we specialize to the case n = 3. We will see that in this dimension any eigenspinorof constant length is a generalized Killing spinor. Proposition 4.10.
Let ψ be a solution of Dψ = Hψ , H ∈ C ∞ ( M ) , of constant length ,on a manifold of dimension n = 3 . Then ψ is a generalized Killing spinor. This proposition is the natural generalization of a result in [30] from n = 2 to n = 3.We will include a simple proof here. Proof.
Let g be the metric on M and h· , ·i the real part of the Hermitian metric on Σ M .We define A ∈ End(
T M ) by g ( A ( X ) , Y ) := h∇ X ψ, Y · ψ i for all X, Y ∈ T M . Note that for any point p ∈ M and any vector X ∈ T p M we have h∇ X ψ, ψ i = ∂ X h ψ, ψ i = 0 , in other words ∇ X ψ ∈ ψ ⊥ = { φ ∈ Σ p M | h φ, ψ i = 0 } . Let e , e , e be an orthonormal basisof T p M . By possibly changing the order of this basis, we can achieve e · e · e = 1 in thesense of endomorphisms of Σ M . The spinors e · ψ , e · ψ and e · ψ form an orthonormalsystem of ψ ⊥ , and because of dim R ψ ⊥ = 3, it is a basis. It follows ∇ X ψ = A ( X ) · ψ .Furthermore h A ( e ) , e i = h∇ e ψ, e · ψ i = h e · ∇ e ψ, e · e · ψ | {z } e · ψ i = H h ψ, e · ψ i | {z } =0 −h e · ∇ e ψ, e · ψ i − h e · ∇ e ψ, e · ψ i = h e · e · ∇ e ψ, ψ i − h∇ e ψ, ψ i = −h e · ∇ e ψ, ψ i = h A ( e ) , e i and similarly h A ( e ) , e i = h A ( e ) , e i and h A ( e ) , e i = h A ( e ) , e i . Thus A is symmetric. (cid:3) Summarizing our knowledge until now, we have:
Corollary 4.11.
Assume that ( M, g ) is a compact connected spin manifold of dimension n = 3 satisfying λ +min ( M, [ g ]) < λ +min ( S ) = (2 π ) / . Then there exist(1) an open, connected and dense subset M ∗ ;(2) a metric g on M ∗ conformal to g | M ∗ and of volume ;(3) an eigenspinor ψ to D g to the real eigenvalue λ ( g ) ,such that ψ has constant length and thus is a generalized Killing spinor on ( M ∗ , g ) . Weobtain a self-adjoint section A of End(
T M ) such that ∇ X ψ = A ( X ) · ψ and tr A = − λ ( g ) . Analytic manifolds.Definition 4.12.
Let g be a Riemannian metric on a smooth manifold M . We saythat [ g ] is an analytic conformal class if M has a compatible structure of a (real-)analyticmanifold for which one of the following equivalent statements holds:(a) there is a (real-)analytic metric h ∈ [ g ];(b) for any point x ∈ M there is an open set U ∋ x , such that there is an analytic metric g U on U with g U ∈ [ g | U ]. Lemma 4.13.
Conditions (a) and (b) in Definition 4.12 are equivalent.Proof.
The implication from (a) to (b) is trivial. The implication from (b) to (a) is a directconsequence of uniformization in dimension n = 2, thus we restrict to the case n ≥ g be a smooth metric in the given analytic conformal class. We have to show thatthe locally defined metrics g U provided by (b) can be deformed conformally such that theymatch together to a globally defined metric. Let L p := { λg p | λ > } ⊂ T ∗ p M ⊗ T ∗ p M , andlet L := S L p . The bundle π : L → M is a smooth R + -principal bundle over M . All localRiemannian metrics g U are local sections of π : L → M , g U : U → L . If two local analytic HE CAUCHY PROBLEMS FOR EINSTEIN METRICS AND PARALLEL SPINORS 19 metrics g U and g ˜ U are given, then there is an analytic function f : ˜ U ∩ U → R + such that g U = f g ˜ U on ˜ U ∩ U . Consequently, π : L → M carries a structure of analytic R + -principalbundle over M , and thus the total space L of the bundle is an analytic manifold. Thesmooth map g : M → L can be approximated in the strong C -topology by an analyticmap g ω : M → L , see [39, Chap. 2, Theorem 5.1] which is proven by Grauert and Remmertin [32]. The map π ◦ g ω : M → M is a smooth analytic map, which is close to the identityin the C -topology, and thus (for suitably chosen g ω ) it is an analytic diffeomorphism.As a consequence, the map g ω ◦ ( π ◦ g ω ) − : M → L is an analytic section of L and thusan analytic representative of the given conformal class. (cid:3) Lemma 4.14.
If an analytic conformal class is conformally flat on a non-empty openset U , and if M is connected, then the conformal class is already conformally flat on M .Proof. Being conformally flat on an open set U is equivalent to the vanishing of the Weylcurvature (resp. Schouten tensor) in dimension m ≥ m = 3). The Weyl curvatureand the Schouten tensor of an analytic metric are analytic as well. Thus if they vanishon U they must vanish on all of M . (cid:3) Lemma 4.15.
Let φ be a smooth solution of Dφ = c | φ | α φ , φ = 0 on a (not necessarilycomplete) analytic Riemannian spin manifold ( U, g, χ ) . Then φ is analytic as well.Proof. The equation is an elliptic semi-linear equation, and has analytic coefficients onthe set M \ φ − (0). We apply analytic regularity results for properly elliptic systems asdeveloped by Douglis and Nirenberg and refined by Morrey, see [48] and [49]. To applythese tools it is convenient to deduce a second order equation D φ = (cid:0) c | φ | α + c grad( | φ | α ) (cid:1) · φ which has again analytic coefficients on M \ φ − (0). The linearization of this second orderequation has the principal symbol of a Laplacian and is thus properly elliptic. The lemmathen follows directly from [49, Theorem 6.8.1] or [48]. (cid:3) Lemma 4.16.
Constant mean curvature hypersurfaces in an analytic Riemannian man-ifolds are analytic. In particular, the metric and the second fundamental form of such ahypersurface are analytic.Proof.
Let M be an n -dimensional hypersurface in an analytic Riemannian manifold ( Z , h )of dimension n + 1. We choose an analytic parametrization U × ( a, b ) → Z with U openin R n , such that locally the hypersurface M is the graph of a function F : U → ( a, b ). Thestandard basis of R n +1 is denoted by e , . . . , e n +1 . The tangent space T ( x,F ( x )) M is thenspanned by ( e i , ∂ i F ), i = 1 , . . . , n .Let h ij ∈ C ω ( U × ( a, b )) be the coefficients of the metric h , and let g ij ∈ C ∞ ( U ) be thecoefficients of g . The inverse matrices are denoted by ( h ij ) ≤ i,j ≤ n +1 and ( g ij ) ≤ i,j ≤ n .The first fundamental form of the hypersurface in the chart given by U is g ij = h ij + h n +1 ,j ∂ i F + h n +1 ,i ∂ j F + h n +1 ,n +1 ( ∂ i F )( ∂ j F ) . The coefficients of the matrices ( g ij ) and ( g ij ) are thus polynomial expressions in h , F and dF . The vector field X := n +1 X i =1 (cid:16)(cid:16) − n X j =1 h ij ∂ j F (cid:17) + h i,n +1 (cid:17) e i is normal to M , and both X and the unit normal vector field ν := X/ | X | h are analyticexpressions in h , F and dF .The second fundamental form has the coefficients k ij = −h∇ ( e i ,∂ i F ) ν, ( e j , ∂ j F ) i = − | X | h h∇ ( e i ,∂ i F ) X, ( e j , ∂ j F ) i = | X | h ( ∂ i ∂ j F + F ( h, dh, F, dF )) , where 1 ≤ i, j ≤ n and F is a polynomial expression in its arguments.The mean curvature H is given as H = n P ij g ij k ij . Thus the mean curvature operator P : F H is a quasi-linear second order differential operator with analytic coefficients.We fix a function ˜ F describing a hypersurface of constant mean curvature, the corre-sponding normal field will be denoted by ˜ X . In other words P ( ˜ F ) is a constant.The linearization ˆ P := T ˜ F P of P in ˜ F is a linear second order differential operator withprincipal symbol R m → R , ξ | ξ | | ˜ X | h . Thus P is (properly) elliptic in a neighborhood of 0.The analytic regularity theorem for elliptic systems of Morrey [49, Theorem 6.8.1] or [48]tells us that ˜ F is analytic, and this implies the lemma. (cid:3) Three-dimensional real projective space.
In this and in the following subsectionwe provide examples of compact Riemannian spin manifolds satisfying (4.5). In the presentsubsection we study deformations of round metrics on R P with a suitable spin structure.This already provides examples of non-analytic Riemannian manifolds with generalizedKilling spinor, showing the necessity of the analyticity assumption in Theorems 1.1 and 2.1.In the following section we will then see that such examples are abundant. Lemma 4.17. If M is a compact spin manifold, we denote the set of metrics with invertibleDirac operator as R inv ( M ) , equipped with the C -topology. Then the function R inv ( M ) → R + , g λ +1 ( g ) is continuous. This lemma is a special case of [13, Prop. 7.1], see also [51, Kor. 1.3.3] for more details.Let us equip SU(2) with the unique bi-invariant metric of sectional curvature 1, henceSU(2) is isometric to S . The left multiplication of SU(2) on itself lifts to an action of SU(2) HE CAUCHY PROBLEMS FOR EINSTEIN METRICS AND PARALLEL SPINORS 21 on Σ SU(2), for any choice of orientation of SU(2) and any choice of the spinor represen-tation. The spinor bundle is then trivialized by left-invariant spinors. A straightforwardcalculation, see e.g. [3], shows that ∇ X φ = ± X · φ for any left-invariant spinor φ and all X ∈ T SU(2). Thus all left-invariant spinors areKilling spinors to the Killing constant ± /
2. The sign depends on the choice of orientationand on the choice of spinor representation. The same discussion also applies to right-invariant spinors, and these are Killing spinors whose Killing constant have the oppositesign. We assume that these choices are made such that left-invariant spinors have Killingconstant − /
2, and thus right-invariant ones have Killing constant +1 / \ SU(2)such that left-invariant spinors on S descend to Γ \ SU(2). Then Γ \ SU(2) carries a complex2-dimensional space of Killing spinors with Killing constant − /
2, but no non-trivial Killingspinor with Killing constant 1 /
2. For quotients SU(2) / Γ, the role of 1 / − / S do not carry any non-trivial Killingspinor.In the special case Γ = {± } both quotients Γ \ SU(2) and SU(2) / Γ are isometric to R P , but they come with different spin structures. These are the two non-equivalent spinstructures on R P . We thus have obtained: Lemma 4.18.
Let σ be the standard metric on the -dimensional real projective space R P . There are two spin structures on R P . For one spin structure Killing spinors tothe constant − / exist, but not for the constant / . For the other spin structure Killingspinors to the constant / exist, but not for the constant − / . Thus for a suitable choice of spin structure, we have λ +min ( R P , [ σ ]) = (cid:0) ω (cid:1) / = π / < λ +min ( S ) = ω / = π / / . Corollary 4.19.
There is a non-analytic conformal class and a spin structure on R P forwhich inequality (4.5) holds.Proof. We choose a metric g close to σ on R P which is conformally flat on some non-empty open set U and non-conformally flat on some non-empty open set U . If g were ananalytic metric, conformal to g , then g would have a vanishing Schouten tensor on U .By Lemma 4.14 it would be flat everywhere, thus also on U . This shows that [ g ] is anon-analytic conformal class. Applying the previous lemmata, we obtain the corollary. (cid:3) The mass endomorphism and application to inequality (4.5).
The goal of thissubsection is to prove that inequality (4.5) holds for “generic” metrics, in a sense explainedbelow.In this section we assume that M is a compact connected spin manifold of dimension n ≥
3, and that the index of M in KO − n ( pt ) vanishes. We fix a point p ∈ M and a flatmetric g flat in a neighborhood U of p , U = M . We assume that U is isometric to a convex ball and that g flat can be extended to a metric on M . The set of all such extensions isdenoted by R U,g flat ( M ). We define R inv U,g flat ( M ) := { g ∈ R U,g flat ( M ) | D g is invertible } , i.e. this is the set of all extensions of g flat such that the Dirac operator is invertible. In [5]we proved that R inv U,g flat ( M ) is open and dense in R U,g flat ( M ) with respect to the C k -topologywhere k ≥ Definition 4.20.
We say that a property (A) holds for generic metrics in R U,g flat ( M ) ifthere is a subset R ′ ⊂ R U,g flat ( M ) that is open and dense with respect to the C k -topologyfor all k ≥
1, such that property (A) holds for all g ∈ R ′ .Using this definition, the above mentioned result from [5] says that the Dirac operatorwith respect to a generic metric is invertible.Given a metric g ∈ R inv U,g flat ( M ), let G be the Green’s function of the Dirac operator on( M, g ) at the point p ∈ M , i.e. a distributional solution of(4.9) DG = δ p Id Σ p M , where δ p is the Dirac distribution at p and G is viewed as a linear map which associates toeach spinor in Σ p M a smooth spinor field on M \ { p } defining a spinor-valued distributionon M . We write G g and D g for G and D to indicate their dependence on the metric g .We also introduce the Euclidean Green’s function centered at 0, defined distributionallyon R n G eucl ψ = − ω n − | x | n x · ψ. It satisfies (4.9) for G = G eucl and D = D eucl on R n .Identifying U with a ball in R n via an isometry, both G = G g and G = G eucl are solutionsof (4.9) on U . Thus D g ( G g − G eucl ) = 0 on U and by elliptic regularity, G g − G eucl is asmooth section, see also [7]. We obtain for any ψ ∈ Σ p M : G g ( x ) ψ = − ω n − | x | n x · ψ + v g ( x ) ψ , where the spinor field v g ( x ) ψ is smooth on U and satisfies D g ( v g ( x ) ψ ) = 0 on U . Definition 4.21.
The mass endomorphism α g : Σ p M → Σ p M for a point p ∈ U ⊂ M isdefined by α g ( ψ ) := v g ( p ) ψ . The mass endomorphism is thus (up to a constant) defined as the zero th order term inthe asymptotic expansion of the Green’s function in Euclidean coordinates around p . Thisdefinition is analogous to the definition of the mass in the Yamabe problem. Theorem 4.22 ([33] for n = 3, [4] for n ≥ . For generic metrics in R U,g flat ( M ) the massendomorphism in p is non-zero. An important application of this theorem is inequality (4.5). The proofs in [7] yield:
HE CAUCHY PROBLEMS FOR EINSTEIN METRICS AND PARALLEL SPINORS 23
Proposition 4.23.
If the mass endomorphism in a point p with flat neighborhood is non-zero, then λ +min ( M, [ g ]) < λ +min ( S n ) . It follows:
Corollary 4.24.
For generic metrics g in R U,g flat ( M ) we have λ +min ( M, [ g ]) < λ +min ( S n ) . We now deduce:
Corollary 4.25.
Let M be an n -dimensional compact spin manifold with vanishing index ind( M ) ∈ KO − n ( pt ) . There there is both an analytic conformal class [ g an ] and a non-analytic, smooth conformal class [ g non − an ] on M with λ +min ( M, [ g an ]) < λ +min ( S n ) , λ +min ( M, [ g non − an ]) < λ +min ( S n ) . In this corollary M is a priori equipped with a C ∞ -structure and the “non-analyticity”means by definition that M does not carry any analytic structure in which g non − an isanalytic. Proof.
We choose an open set U and a metric g flat as above. Then choose g ∈ R U,g flat ( M )with λ +min ( M, [ g ]) < λ +min ( S n ). Choose another smooth metric g non − an , coinciding on U with g = g flat , such that g non − an is not (everywhere) conformally flat on M \ U , and C -closeenough to g so that λ +min ( M, [ g non − an ]) < λ +min ( S n ). The metric g non − an is conformally flaton U but not on M \ U , hence its Schouten tensor cannot be analytic in any analyticstructure. Thus as in Lemma 4.14 the conformal class [ g non − an ] cannot be analytic.At the same time, g can be C -approximated by an analytic metric g an so that theinequality λ +min ( M, [ g an ]) < λ +min ( S n ) continues to hold. Such an analytic approximationcan be done either with Abresch’s smoothing technique or by using the Ricci flow: if g t isa solution of the Ricci flow equation ddt ( g t ) = − g t , defined for short times t ∈ [0 , t )with initial data g = g , then g t is analytic for all t >
0. We set g an := g t for a sufficientlysmall t > (cid:3) Analytic examples.
Summarizing the results of the preceding subsections we obtain.
Theorem 4.26.
Assume that ( M, [ g an ]) is a compact connected analytic Riemannian spinmanifold of dimension with λ +min ( M, [ g an ]) < λ +min ( S n ) . Then there is a connected, openand dense subset M ∗ of M carrying an analytic metric g ∗ and an analytic spinor field ψ ∈ Γ(Σ g ∗ M ∗ ) such that(1) g ∗ is conformal to g an | M ∗ ;(2) vol( M ∗ , g ∗ ) = 1 ;(3) ψ is a generalized-Killing spinor on ( M ∗ , g ∗ ) . Such Riemannian metrics g an exist on each compact 3-dimensional spin manifold, dueto the preceding section. The corresponding endomorphism W is then analytic as well,and Theorem 1.1 can be applied. We obtain a Ricci-flat metric of the form dt + g t where g = g ∗ defined on an open neighborhood of { } × M ∗ in R × M ∗ , and carrying a parallel spinor. Further the mean curvature of { } × M ∗ in this neighborhood is constant and equalto (2 / λ +min ( M, [ g ]).4.8. Non-analytic examples.
Here we finally prove the existence of metrics with gener-alized Killing spinors on manifolds with non-analytic metrics. According to Lemma 4.16such manifolds do not embed isometrically as constant mean curvature hypersurfaces inRicci-flat manifolds. Thus the analyticity assumptions in Theorem 1.1 cannot be removed.
Theorem 4.27.
Any -dimensional compact connected spin manifold M with a fixed C ∞ -structure has a connected open dense subset M ∗ carrying a smooth Riemannian metric g ∗ with a generalized Killing spinor, such that the metric is not analytic for any choice ofanalytic structure on M . For this manifold ( M ∗ , g ∗ ) the associated formal solution providedby Proposition 2.6 cannot be chosen to be Ricci-flat on a neighborhood of { } × M , in otherwords the conclusion of Theorem 1.1 does not hold. If M = R P or more generally if M = Γ \ SU(2) for a non-trivial subgroup Γ of SU(2) , then we can find such a Riemannianmetric g ∗ defined on the whole manifold M .Proof. By Corollary 4.25 there exists a smooth conformal class [ g non − an ] whose Schoutentensor vanishes on a non-empty open set and does not vanish on another open set, andfor which λ +min ( M, [ g non − an ]) < λ +min ( S ). The infimum in (4.1) is then attained, accordingto Theorem 4.4, at a generalized metric g ∗ , which is a smooth Riemannian metric on aconnected dense open subset M ∗ of M . It is clear that the restricted conformal class[ g non − an | M ∗ ] is not analytic, and thus the metric g ∗ cannot be analytic either. Theorem 4.4provides moreover a Dirac eigenspinor of constant length on ( M ∗ , g ∗ ) which is, due toSubsection 4.3, a generalized Killing spinor. Furthermore, the trace of the associatedsymmetric tensor W ∈ End(
T M ) is constant and equal to − (2 / λ +min ( M, [ g non − an ]).If the formal solution provided by Proposition 2.6 (for M ∗ instead of M ) were Ricci-flat in a neighborhood of { } × M ∗ , then M ∗ would be a hypersurface of constant meancurvature in a 4-dimensional Ricci-flat manifold. As Ricci-flat metrics are analytic in asuitable analytic structure, Lemma 4.16 would imply that g ∗ was analytic, which is acontradiction.Now assume that M = R P . We take a sequence of non-analytic metrics g i (constructedsimilarly as above) converging in the C ∞ -topology to the standard round metric σ on R P . As the functional F gq depends continuously on g in the C -topology, we see for q = 2 n/ ( n + 1)(4.10) µ := lim inf i →∞ µ q ( R P , g i ) ≥ µ q ( R P , σ ) = (cid:16) ω (cid:17) / > µ q ( S ) = (cid:16) ω (cid:17) / Now let ψ i be a maximizing spinor on ( R P , g i ) with L p -norm 1, p = 2 n/ ( n −
1) = 3.These ψ i are uniformly bounded in the C -norm. This uniform C -boundedness followsfrom [2, Theorem 6.1] whose proof is also valid for p = 2 n/ ( n −
1) although the formulationof [2, Theorem 6.1] assumed p < n/ ( n − ψ i is abounded sequence in C ,α for any α ∈ (0 , HE CAUCHY PROBLEMS FOR EINSTEIN METRICS AND PARALLEL SPINORS 25
After passing to a suitable subsequence we then see that ψ i converges to a solution ψ of D σ ψ = µ − | ψ | ψ, k ψ k L ( R P ,σ ) = 1 . Calculating F n/ ( n +1) ( ψ ) = ¯ µ we conclude µ = µ q ( R P , σ ). Using the regularity theorem[2, Prop. 5.1] one sees that ψ is C . We now apply [2, Prop. 4.1] where ψ ∈ Γ(Σ S ) is thepullback of ψ to S . One calculates F n/ ( n +1) ( ψ ) = µ q ( S ), thus ψ is a maximizing spinoron S . The conformal map A : S → S in the conclusion of [2, Prop. 4.1] has to be anisometry as it is the lift of a map R P → R P . Thus [2, Prop. 4.1] implies that ψ is aKilling spinor to the Killing constant − /
2. As such a Killing spinor nowhere vanishes, ψ i nowhere vanishes for large i .The other quotients Γ \ SU(2) are completely analogous. (cid:3)
Using products with manifolds carrying parallel spinors, one can easily obtain in everydimension n ≥ n -dimensional manifolds with generalized Killing spinorswhich do not embed isometrically as hypersurfaces in manifolds with parallel spinors.More precisely we have the following: Lemma 4.28.
Let ( M ∗ , g ∗ ) be a 3-dimensional non-analytic Riemannian manifold withgeneralized Killing spinors given by Theorem 4.27. Then the Riemannian product ( M ∗ , g ∗ ) × ( R n − , g eucl ) carries a generalized Killing spinor Ψ but can not be embedded isometrically as a hypersur-face in any manifold with parallel spinors which restrict to Ψ .Proof. Let p ∗ (Σ M ∗ ) and p ∗ (Σ R n − ) denote the pullbacks to Z := M ∗ × R n − of the spinbundles of ( M ∗ , g ∗ ) and ( R n − , g eucl ) with respect to the standard projections. It is astandard fact that the spin bundle Σ Z is isomorphic to p ∗ (Σ M ∗ ) ⊗ p ∗ (Σ R n − ) if n isodd and to p ∗ (Σ M ∗ ) ⊗ p ∗ (Σ R n − ) ⊗ C if n is even, and this isomorphism preserves thespin connections. The isomorphism can be chosen such that in the first case, the Cliffordproduct is given by ( X , X ) · ( φ ⊗ ψ ) = ( X · φ ) ⊗ ψ + φ ⊗ ( X · ω C · ψ ) , where ω C is the complex volume form in the Clifford algebra of R n − . In the second case,the Clifford product is given by( X , X ) · ( φ ⊗ ψ ⊗ v ) = ( X · φ ) ⊗ ψ ⊗ a ( v ) + φ ⊗ ( X · ψ ) ⊗ b ( v ) , for every v ∈ C , where a = (cid:18) − (cid:19) and b = (cid:18) (cid:19) . The first assertion now followsimmediately: take any generalized Killing spinor φ on M ∗ satisfying ∇ X φ = W ( X ) · φ forall X ∈ T M ∗ and let ψ be a parallel spinor on R n − . One can of course assume that ω C · ψ = ψ if n is odd. Then Ψ := φ ⊗ ψ (resp. Ψ := φ ⊗ ψ ⊗ (cid:18) (cid:19) ) is a generalized Killingspinor on Z for n odd (resp. even), with associated tensor ¯ W = (cid:18) W
00 0 (cid:19) . To prove the second assertion, assume that Z is a hypersurface in some spin manifold¯ Z and that Φ is a parallel spinor on ¯ Z restricting to Ψ on Z . The second fundamentalform of Z is ¯ W , which has constant trace by construction. Thus Z has constant meancurvature, so is analytic by Lemma 4.16. Each factor of Z is then analytic, contradictingthe non-analyticity of M ∗ . (cid:3) References
1. B. Ammann,
A spin-conformal lower bound of the first positive Dirac eigenvalue , Diff. Geom. Appl. (2003), 21–32.2. , The smallest Dirac eigenvalue in a spin-conformal class and cmc-immersions , Comm. Anal.Geom. (2009), 429–479.3. , A variational problem in conformal spin geometry , Habilitationsschrift, Universit¨at Hamburg,2003.4. B. Ammann, M. Dahl, A. Hermann, E. Humbert,
Mass endomorphism, surgery and perturbations ,arXiv:1009.5618, to appear in Ann. Inst. Fourier.5. B. Ammann, M. Dahl, E. Humbert,
Harmonic spinors and local deformations of the metric , Math. Res.Letters (2011), 927–936.6. B. Ammann, J.-F. Grosjean, E. Humbert, B. Morel, A spinorial analogue of Aubin’s inequality , Math.Z. (2008), 127–151.7. B. Ammann, E. Humbert, B. Morel,
Mass endomorphism and spinorial Yamabe type problems , Comm.Anal. Geom. (2006), 163–182.8. B. Ammann, P. Jammes, The supremum of conformally covariant eigenvalues in a conformal class ,Proc. Conf. Leeds on the occasion of J. Woods 60 th birthday, 2009, Variational Problems in DifferentialGeometry (R. Bielawski, K. Houston, and M. Speight, eds.), London Math. Soc. Lect. Notes Ser. ,(2011), 1–23.9. M. T. Anderson, M. Herzlich, Unique continuation results for Ricci curvature and applications,
J. Geom.Phys. (2008), no. 2, 179–207.10. T. Aubin, ´Equations diff´erentielles non lin´eaires et probl`eme de Yamabe concernant la courburescalaire , J. Math. Pur. Appl. IX. Ser. (1976), 269–296.11. H. Baum, Th. Friedrich, R. Grunewald, I. Kath, Twistor and Killing Spinors on Riemannian Mani-folds,
Teubner-Verlag, Stuttgart-Leipzig, 1991.12. C. B¨ar,
Real Killing spinors and holonomy , Commun. Math. Phys. (1993), 509–521.13. ,
Metrics with harmonic spinors , Geom. Funct. Anal. (1996), 899–942.14. , Zero sets of solutions to semilinear elliptic systems of first order , Invent. Math. (1999),no. 1, 183–202.15. C. B¨ar, P. Gauduchon, A. Moroianu,
Generalized Cylinders in Semi-Riemannian and Spin Geometry,
Math. Z. (2005), 545–580.16. A. Besse,
Einstein manifolds , Ergebnisse der Mathematik und ihrer Grenzgebiete (3) , Springer-Verlag, Berlin, 1987.17. O. Biquard, Continuation unique `a partir de l’infini conforme pour les m´etriques d’Einstein,
Math.Res. Lett. (2008), no. 6, 1091–1099.18. B. Booss-Bavnbek, M. Marcolli, B.-L. Wang, Weak UCP and perturbed monopole equations , Internat.J. Math. (2002), no. 9, 987–1008.19. J. P. Bourguignon, P. Gauduchon, Spineurs, op´erateurs de Dirac et variations de m´etriques , Commun.Math. Phys. (1992), 581–599.20. R. Bryant,
Non-embedding and non-extension results in special holonomy , The many facets of geometry– A tribute to Nigel Hitchin, 346–367, Oxford Univ. Press, Oxford, 2010.21. D. Conti,
Embedding into manifolds with torsion , Math. Z. (2011), no. 3-4, 725–751.
HE CAUCHY PROBLEMS FOR EINSTEIN METRICS AND PARALLEL SPINORS 27
22. D. Conti, S. Salamon,
Reduced holonomy, hypersurfaces and extensions , Int. J. Geom. Methods Mod.Phys. (2006), no. 5-6, 899–912.23. , Generalized Killing spinors in dimension 5 , Trans. Amer. Math. Soc. (2007), no. 11,5319–5343.24. J. Dieudonn´e, ´El´ements d’analyse. Tome IV: Chapitres XVIII `a XX,
Cahiers Scientifiques, Fasc.XXXIV, Gauthier-Villars, 1971.25. D. DeTurck,
The Cauchy problem for Lorentz metrics with prescribed Ricci curvature , CompositioMath. (1983), no. 3, 327–349.26. , Existence of metrics with prescribed Ricci curvature: local theory,
Invent. Math. (1981/82),no. 1, 179–207.27. D. DeTurck, J. Kazdan, Some regularity theorems in Riemannian geometry,
Ann. Sci. ´Ecole Norm.Sup. (4) (1981), no. 3, 249–260.28. C. L. Fefferman, C. R. Graham, Conformal invariants,
Asterisque, vol. hors s´erie, Soc. Math. France,(1985), 95–116.29. Y. Four`es-Bruhat,
R´esolution du probl`eme de Cauchy pour des ´equations hyperboliques du secondordre non lin´eaires,
Bull. Soc. Math. France (1953), 225–288.30. Th. Friedrich, On the spinor representation of surfaces in Euclidean -space , J. Geom. Phys. (1998),143–157.31. M. Golubitsky, V. Guillemin, Stable mappings and their singularities,
Graduate Texts in Mathematics , Springer-Verlag, New York-Heidelberg, 1973.32. H. Grauert, On Levi’s problem and the imbedding of real-analytic manifolds , Ann. Math. (2) (1958), 460–472.33. A. Hermann, Generic metrics and the mass endomorphism on spin -manifolds , Ann. Global Anal.Geom. (2010), 163–171.34. A. Hermann, Dirac eigenspinors for generic metrics , PhD thesis, Universit¨at Regensburg,arXiv:1201.5771.35. A. Hermann,
Zero sets of eigenspinors for generic metrics , arXiv:1208.1414.36. O. Hijazi,
A conformal lower bound for the smallest eigenvalue of the Dirac operator and Killingspinors , Commun. Math. Phys. (1986), 151–162.37. ,
Premi`ere valeur propre de l’op´erateur de Dirac et nombre de Yamabe , C. R. Acad. Sci. Paris (1991), 865–868.38. ,
Spectral properties of the Dirac operator and geometrical structures , Ocampo, Hernan (ed.),Geometric methods for quantum field theory. Proc. summer school, Villa de Leyva, Colombia, 1999,World Scientific, Singapore, 116–169, 2001.39. M. W. Hirsch,
Differential topology , Graduate Texts in Mathematics, , Springer-Verlag, 1976.40. N. Hitchin, Stable forms and special metrics , Global differential geometry: the mathematical legacyof Alfred Gray (Bilbao, 2000), 70–89, Contemp. Math. , Amer. Math. Soc. Providence, RI, 2001.41. E. C. Kim,
A local existence theorem for the Einstein-Dirac equation , J. Geom. Phys. , (2002),376–405.42. N. Koiso, Hypersurfaces of Einstein manifolds . Ann. Sci. ´Ecole Norm. Sup. (1981), no. 4, 433–443.43. M.-A. Lawn, J. Roth, Isometric immersions of hypersurfaces in 4-dimensional manifolds via spinors ,Diff. Geom. Appl. (2010), no. 2, 205–219.44. J. M. Lee, T. H. Parker, The Yamabe problem , Bull. Amer. Math. Soc. (1987), 37–91.45. J. Lott, Eigenvalue bounds for the Dirac operator , Pacific J. Math. (1986), 117–126.46. B. Morel,
The energy-momentum tensor as a second fundamental form , math.DG/0302205.47. B. Morel,
Surfaces in S and H via spinors , Actes de S´eminaire de Th´eorie Spectrale et G´eom´etrie, , Ann´e 2004–2005, 131–144, Univ. Grenoble I, Saint-Martin-d’H`eres, 2005.48. C. B. Morrey, Jr., On the analyticity of the solutions of analytic non-linear elliptic systems of partialdifferential equations. I. Analyticity in the interior,
Amer. J. Math. (1958), 198–218.
49. C. B. Morrey, Jr.,
Multiple integrals in the calculus of variations , Die Grundlehren der mathematischenWissenschaften, , Springer Verlag, New York, 1966.50. M. Obata,
The conjectures on conformal transformations of Riemannian manifolds , J. Differ. Geom. (1971/72), 247–258.51. F. Pf¨affle, Eigenwertkonvergenz f¨ur Dirac-Operatoren , Ph.D. thesis, University of Hamburg, Germany,2002, Shaker Verlag, Aachen, 2003.52. R. Schoen,
Conformal deformation of a Riemannian metric to constant scalar curvature , J. Differ.Geom. (1984), 479–495.53. R. Schoen, S.-T. Yau, Conformally flat manifolds, Kleinian groups and scalar curvature , Invent. Math. (1988), 47–71. Bernd Ammann, Fakult¨at f¨ur Mathematik, Universit¨at Regensburg, 93040 Regensburg,Germany
E-mail address : [email protected] Andrei Moroianu, Universit´e de Versailles-St Quentin, Laboratoire de Math´ematiques,UMR 8100 du CNRS, 45 avenue des ´Etats-Unis, 78035 Versailles, France
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