A SPINORIAL ENERGY FUNCTIONAL: CRITICAL POINTS AND GRADIENT FLOW

BERND AMMANN, HARTMUT WEISS, AND FREDERIK WITT

Abstract. Let M be a compact spin manifold. On the universal bundle of unit spinors we study a natural energy functional whose critical points, if dim M 3, are precisely the pairs g, φ consisting of a Ricci-flat Riemannian metric g together with a parallel g-spinor φ. We investigate the basic properties of this functional and study its negative gradient flow, the so-called spinor flow. In particular, we prove short-time existence and uniqueness for this flow.

1. Introduction

Approaching special holonomy metrics and related structures from a spinorial point of view often gives interesting insights. For example, one can characterise Ricci-flat metrics of special holonomy in terms of parallel spinors [26], [39]. This enabled Wang to show that if M, g is a simply-connected, irreducible Riemannian manifold with a parallel spinor, any metric in a suitable Einstein neighbourhood of that metric must also admit a parallel spinor and is thus of special holonomy as well, see [40]. More generally, one can consider Killing spinors (in the sense of [7]) which forces the underlying metric g to be Einstein. In fact, they arise in connection with Einstein-Sasaki structures (see for instance [7], [10]) and also relate to Gray’s notion of weak holonomy groups [23]. Bär’s classification [5] links Killing spinors to parallel spinors on the cone M ℝ_{}with warped product metric t^{2}g dt^{2}, thereby relating these Einstein geometries to special holonomy metrics. More generally still, one can consider generalised Killing spinors in the sense of [6] in connection with embedding problems. In [1] it is shown that a manifold of arbitrary dimension n which carries a real-analytic generalised Killing spinor embeds isometrically into a manifold of dimension n 1 with a parallel spinor. This generalises results on Hitchin’s embedding problem [27] in dimension n 7, n 6 and n 5 with co-calibrated G_{2}-, half-flat and hypo-structures respectively [12] to arbitrary dimensions.

The present article is the first one of a programme to understand and solve these spinor field equations from a variational point of view. This also puts a previous result of the last two authors on a certain heat flow on 7-dimensional manifolds [42] into a framework which is valid in any dimension. More concretely, let M be an n-dimensional, compact, oriented spin manifold with a given spin structure, and consider the universal bundle of unit spinors S ΣM M. A section Φ Γ S ΣM can be regarded as a pair g, φ where g is a Riemannian metric and φ Γ Σ_{g}M is a g-spinor of constant length one. We then introduce the energy functional defined by

ℝ_{0}, Φ \frac{1}{2} _{M} ^{g}φ _{g}^{2} dv^{g},

where ^{g} denotes the Levi-Civita connection on the g-spinor bundle Σ_{g}M, _{g} the pointwise norm on T^{}M Σ_{g}M and dv^{g} the Riemann-Lebesgue measure given by the volume form of g. If M is a compact Riemann surface, then a conformal immersion F M ℝ^{3} induces a spin structure on M and a section g, φ on M. The spinorial version of the Weierstrass representation (see [18]), implies D_{g}φ Hφ where H is the mean curvature function of the immersed surface. In this case we obtain 2 g, φ _{M} H^{2} dv^{g} which is the Willmore energy of the immersion. Our functional thus extends the Willmore functional to a larger domain. We elaborate on this point in [2].

Our first result characterises the (unconstrained) critical points.

Proposition A (Critical points). Let n 3. Then g, φ is critical if and only if it is an absolute minimiser, i.e. ^{g}φ 0. In particular, the metric g is Ricci-flat and of special holonomy.

The case dim M 2 is of a different flavour and is be also treated in detail in [2]. There, we have a trichotomy for the absolute minimisers according to the genus γ, namely twistor spinors for γ 0, parallel spinors for γ 1 and harmonic spinors for γ 2. However, non-minimal critical points do exist for γ 1. Coming back to the case of n 3, we can also consider critical points subject to the constraint Vol g _{M} dv^{g} 1. Particular solutions are given by Killing spinors (as above in the sense of [7]). We expect more general solutions to exist, but we will leave this, as well as a systematic investigation of the resulting “soliton equation” following the lines of [43], to a further paper. Proposition A as well as the related results we have just mentioned follow from the computation of the negative L^{2}-gradient Q T of . The main technical ingredient is the Bourguignon–Gauduchon “partial connection” on the fiber bundle S ΣM which yields a horizontal distribution on the (Fréchet) vector bundle , where is the space of Riemannian metrics on M, cf. [9]. In T_{g,φ} the vertical space is Γ φ^{}, the space of spinors pointwise orthogonal to φ. The Bourguignon–Gauduchon horizontal distribution provides a natural complement, isomorphic to T_{g} , the space of symmetric 2-forms on M. This formalism also underlies Wang’s pioneering work on the deformation of parallel spinors under variation of the metric [40] mentioned above. However, instead of using the universal spinor bundle, he considers a fixed spinor bundle where the variation of the metric materialises as a variation of the connections with respect to which one differentiates the spinor fields. Bearing this in mind, some of our formulæ already appear in [40], see Remark 4.13.

In order to detect critical points it is natural to consider the negative gradient flow

\frac{∂}{∂t}Φ_{t} Q Φ_{t} , Φ_{0} Φ  (1)

for an arbitrary initial condition Φ g, φ . One may wonder if any favourable behaviour of this flow is to be expected from convexity properties of the functional. However, the functional is convex only under variation of the spinor field φ alone (keeping the metric g fixed). In general, under variation of both the spinor field and the metric, convexity fails as is shown by the existence of critical points on the 2-torus with indefinite Hessian [2]. Nevertheless, and somewhat suprisingly, the second order partial differential operator defined by linearising the gradient of the energy has positive definite principal symbol up to diffeomorphisms. From there we obtain the

Theorem B (Short-time existence and uniqueness). For all Φ , there exists ε 0 and a smooth family Φ_{t} for t 0, ε such that (1) holds. Furthermore, if Φ_{t} and Φ_{t}^{}are solutions to (1), then Φ_{t} Φ_{t}^{}whenever defined. Hence Φ_{t} is uniquely defined on a maximal time-interval 0, T for some 0 T .

The group of spin-diffeomorphisms { \atop Diff}_{s} M acts on pairs g, φ in a natural way and contains in particular the universal covering group { \atop Diff}_{0} M of the group of diffeomorphisms isotopic to the identity. Since the operator Q is equivariant with respect to this action, its linearisation has an infinite-dimensional kernel. The principal symbol of the linearisation is positive semi-definite. A key observation is that the kernel of the principal symbol is solely due to this diffeomorphism invariance. Though the standard theory of (strongly) parabolic evolution equations does not apply directly anymore, Q is still weakly parabolic. Using a variant of DeTurck’s trick [14] as in [42], we can prove Theorem B for the perturbed flow equation with

{ \atop Q_{Φ_{0}}} T , Φ Q Φ λ_{Φ}^{}X_{Φ_{0}} Φ

instead of Q. Here, X_{Φ_{0}} is a certain vector field which depends on the 1-jet of the initial condition Φ_{0}. Further, λ_{Φ}^{}X 2δ_{g}^{}X^{}, _{X}^{g}φ \frac{1}{4}dX^{}φ where Φ g, φ and δ_{g}^{}is the adjoint of the divergence operator associated with g. The second component of λ_{Φ}^{}is the spinorial Lie derivative as defined in [9] and [29]. Roughly speaking the degeneracy of Q is eliminated by breaking the Diff_{0} M - equivariance with an additional Lie derivative term. One can then revert solutions for the perturbed flow back into solutions of (1).

The perturbed operator Q̃_{Φ_{0}} also appears in connection with the premoduli space of critical points. The set of critical points of , Crit , fibres over the subset of Ricci-flat metrics with a parallel spinor, the fibres being the finite-dimensional vector spaces of parallel spinors. In general, the dimension of these spaces need not to be locally constant. This, however, will be the case if M is simply-connected and g irreducible as a consequence of Wang’s stability [40] and Goto’s unobstructedness [22] theorem. Recall that a function is said to be Morse-Bott if its critical set is smooth and if it is non-degenerate transverse to the critical set. We get

Theorem C (Smoothness of the critical set). Let M be simply-connected and Φ̄ ḡ, φ̄ be an irreducible critical point, i.e. its underlying metric is irreducible. Then Crit is smooth at Φ̄, i.e. its Zariski tangent space is integrable. Further, Q̃ 1 ; Φ̄ 0 is a slice for the { \atop Diff}_{0} M -action on Crit , that is, the premoduli space of parallel spinors Crit { \atop Diff}_{0} M is smooth at Φ̄. If all critical points are irreducible, then is Morse-Bott.

Note that under the assumption that M is simply-connected any critical point is irreducible in dimensions 4, 6 and 7. The same holds in dimension 8 unless M is a product of two K3-surfaces. Theorem C holds more generally on certain non simplyconnected manifolds, see Theorem 4.17. In fact, ongoing work [3] will actually show that one can remove the assumptions “simply-connected” and “irreducible” so that Crit is always smooth, cf. also Remark 4.23.

Theorem C and the formula for the second variation of which we compute in Section 4.3 give all the necessary ingredients for stability of the flow in the sense of [42, Theorem 8.1]. Namely, in a suitable C^{}-neighbourhood of an irreducible critical point, we expect the flow (1) to exist for all times and to converge modulo diffeomorphisms to a critical point. This is now work in progress by Lothar Schiemanowski as part of his thesis project. Ideally, the flow could become a tool for detecting special holonomy metrics, metrics with (generalised) Killing spinors, twistor spinors and other solutions to “natural” spinor field equations. Also the limit spaces when solutions of the flow develop singularities should be of interest and we hope to develop these issues further in the near future.

Acknowledgements. The authors thank the referees for carefully reading the manuscript which led to considerable improvements of the text.

2. Spin geometry

In this section we set up our conventions relevant for the subsequent computations and recall the basic spin geometric definitions. Suitable references for this material are [19] and [31].

Throughout this paper, M^{n} will denote a connected, compact, oriented smooth manifold of dimension n 2. We write for the space of Riemannian metrics on M. The choice of g gives us the Riemannian volume form vol_{g} and allows us to identify vectors with covectors via the musical isomorphisms ^{}T M T^{}M and ^{}T^{}M T M. We will often drop the disctinction between v and v^{}for v T M (resp. between ξ and ξ^{}for ξ T^{}M). In particular, we will identify a local orthonormal frame e_{1}, …, e_{n} with its dual coframe. Further, the metric g induces bundle metrics on the tensor powers ^{p} T^{}M and the exterior powers Λ^{p}T^{}M: If e_{1}, …, e_{n} is a local orthonormal frame for T M, then e_{i_{1}} … e_{i_{p}}, 1 i_{1}, …, i_{p} n is a local orthonormal frame for ^{p} T^{}M and e_{1_{1}} … e_{i_{p}}, 1 i_{1} … i_{p} n for Λ^{p}T^{}M. We embed Λ^{p}T^{}M into ^{p} T^{}M via

ξ_{1} … ξ_{n} _{σ S_{p}} sgn σ ξ_{σ 1} … ξ_{σ p},

which, however, is not an isometric embedding. For example, considered as an element in Λ^{n}T^{}M, the volume form is given by vol_{g} e_{1} … e_{n} with respect to a local orthonormal frame and has unit length. Considered as an anti-symmetric element in ^{n} T^{}M it has length \overline{n!} and satisfies vol_{g} e_{1}, …, e_{n} 1. In the symmetric case, we only consider ^{2} T^{}M which we view as a subspace of ^{2} T^{}M via the embedding

ξ_{1} ξ_{2} \frac{1}{2} ξ_{1} ξ_{2} ξ_{2} ξ_{1} .

We then equip ^{2} T^{}M with the induced metric. Using this convention, the metric may be expressed as g _{i 1}^{n} e_{i} e_{i} _{i 1}^{n} e_{i} e_{i} with respect to a local orthonormal frame and has length \overline{n}. In all cases we generically write , _{g} for metrics on tensors associated with g in this way, and _{g} for the associated norm. Finally, we denote the associated Riemann–Lebesgue measure by dv^{g}. This yields an L^{2}-inner product on any such bundle via

α, β _{g} _{M} α, β _{g} dv^{g}

for α, β ^{p} T^{}M, resp. Λ^{p}T^{}M. We write _{g} for the associated L^{2}-norm.

In the sequel we require M to be a spin manifold. By definition, this means that for the principal GL_{n}^{}-bundle of oriented frames P there exists a twofold covering by a principal { \atop GL}_{n}^{}-bundle P̃ which fiberwise restricts to the universal covering map θ { \atop GL}_{n}^{}GL_{n}^{}for n 3 (resp. the connected double covering for n 2). We call P̃ P a spin structure. We always think of a spin manifold as being equipped with a fixed spin structure which in general is not unique. In fact, H^{1} M, ℤ_{2} acts freely and transitively on the set of equivalence classes of spin structures. The additional choice of g reduces P to the principal SO_{n}-fiber bundle P_{g} of oriented gorthonormal frames. This in turn is covered by a uniquely determined principal Spin_{n}-bundle P̃_{g} which reduces P̃, where Spin_{n} is by definition the inverse image of SO_{n} under θ, i.e. Spin_{n} θ^{1} SO_{n} .

To introduce spinors we need to consider representations of Spin_{n}. For simplicity we will restrict the discussion in this article to complex representations unless specified otherwise. However, all results continue to hold if we replace complex representations by real representations taking into account the real representation theory, cf. for instance [31, Proposition I.5.12] and the discussion in Section 4.1.3.

If we view Spin_{n} as a subgroup of the group of invertible elements in Cliff_{n}, the Clifford algebra of Euclidean space ℝ^{n}, g_{0} , then every irreducible representation of Cliff_{n} restricts to a representation of Spin_{n}. As shown in [31, I.5] there is up to isomorphism only one such representation of Spin_{n} in any dimension, which is called the spin representation Σ_{n}. Note that if n is odd this representation extends to two non-isomorphic representations of the Clifford algebra which only differ by a sign. The non-trivial element in the kernel of Spin_{n} SO_{n} acts as id in this spin representation. The elements of Σ_{n} are referred to as spinors. The (complex) dimension of Σ_{n} is 2^{n 2}, where k r is the largest integer k r. For n even, Σ_{n} is an irreducible representation of the Clifford algebra, but decomposes as a Spin_{n} representation into Σ_{n} Σ_{n}^{}Σ_{n}^{}. For n odd, Σ_{n} is an irreducible representation of Spin_{n}, which however extends to two non-equivalent representations of Cliff_{n}. An important feature of spinors is that they can be multiplied by vectors. More precisely, the above mentioned representation of the Clifford algebra defines a Spin_{n}-equivariant bilinear map μ ℝ^{n} Σ_{n}^{}Σ_{n}^{}called Clifford multiplication. We usually write v φ μ v, φ . This can be extended to forms as follows: If φ Σ_{n}, α Λ^{p}ℝ^{n} and E_{1}, …, E_{n} denotes the standard oriented orthonormal basis of ℝ^{n}, then

α φ _{1 j_{1} ... j_{p} n} α E_{j_{1}}, …, E_{j_{p}} E_{j_{1}} … E_{j_{p}} φ,

which is again a Spin_{n}-equivariant operation. In particular, we have

X Y φ X Y φ g X, Y φ.

Finally, there exists a Spin_{n}-invariant hermitian inner product h on Σ_{n} which for us means in particular that h is positive definite. This gives rise to a positive definite real inner product , Re h for which Clifford multiplication is skewadjoint, that is, v φ, ψ φ, v ψ for all v ℝ^{n}, φ, ψ Σ_{n}. It follows that α φ, ψ 1 ^{p p 1 2} φ, α ψ for α Λ^{p}ℝ^{n}.

Coming back to the global situation, the choice of a metric enables us to define the vector bundle

Σ_{g}M P̃_{g} _{Spin_{n}} Σ_{n}

associated with P̃_{g}. By equivariance, Clifford multiplication, the hermitian inner product and the real inner product , make global sense on M and will be denoted by the same symbols. We denote by

g Γ Σ_{g}M and _{g} φ _{g} φ _{g} 1

the space of (unit) sections, called (unit) spinor fields, or (unit) spinors for short. These spaces carry an L^{2}-inner product given by

φ, ψ _{g} _{M} φ, ψ dv^{g}.

This inner product is again a real inner product. It is in fact the real part of an hermitian inner product, but for our purposes it is more convenient to work with the real inner product. The Levi-Civita connection ^{g} can be lifted to a metric connection on Σ_{g}M which we denote by ^{g} as well. For X, Y Γ T M and φ _{g} it satisfies

_{X}^{g} Y φ _{X}^{g}Y φ Y _{X}^{g}φ.

In terms of a local representation b̃, φ̃ of φ, where for an open U M, φ̃ is a map U Σ_{n} and b̃ U P̃_{g} covers a local orthonormal basis b e_{1}, …, e_{n} U P_{g}, we have

_{X}^{g}φ b̃, dφ̃ X \frac{1}{2} _{i j} g _{X}^{g}e_{i}, e_{j} E_{i} E_{j} φ̃ .  (2)

The action of the curvature operator associated with the Levi-Civita connection, R^{g} X, Y T M T M for vector fields X, Y Γ T M , gives also rise to an action g _{g}, namely

R^{g} X, Y φ _{X}^{g} _{Y}^{g} φ _{Y}^{g} _{X}^{g}φ _{X,Y}^{g}φ.  (3)

Expressed in a local orthonormal basis we have

R^{g} X, Y φ \frac{1}{2} _{j k} g R^{g} X, Y e_{j}, e_{k} e_{j} e_{k} φ

which yields using the first Bianchi identity

_{i 1}^{n} e_{i} R^{g} X, e_{i} φ \frac{1}{2}Ric X φ.

In particular, if g admits a parallel spinor, i.e. there is a φ _{g} 0 satisfying gφ 0, then g is Ricci-flat.

3. The dependence of the spinors on the metric

The very definition of spinors requires the a priori choice of a metric: Any finitedimensional representation of { \atop GL}_{n}^{}factorises via GL_{n}^{}, so that there are no spin representations for the general linear group. Consequently, any object involving spinors will in general depend on the metric. We therefore need a way to compare spinors defined with respect to different metrics, that is, we need some kind of connection.

3.1. The universal spinor bundle. As a bundle associated with the GL_{n}^{}-principal bundle P of oriented frames, the bundle of positive definite bilinear forms is given by

_{}^{2} T^{}M P _{GL}_{n}^{}GL_{n}^{}SO_{n} P SO_{n}.

In particular, the projection P _{}^{2}T^{}M is a principal SO_{n}-bundle. If a spin structure P̃ P is chosen, we may also write

_{}^{2} T^{}M P̃ { \atop GL}_{n}^{}GL_{n}^{}SO_{n} P̃ { \atop GL}_{n}^{}{ \atop GL}_{n}^{}Spin_{n} P̃ Spin_{n}

and the projection P̃ _{}^{2}T^{}M becomes a Spin_{n}-principal bundle. The universal spinor bundle is then defined as the associated vector bundle

π ΣM P̃ _{Spin_{n}} Σ_{n} _{}^{2}T^{}M

where Σ_{n} is the n-dimensional spin representation of Spin_{n}. Composing π with the fiber bundle projection _{}^{2}T^{}M M one can view ΣM also as a fiber bundle over M with fiber { \atop GL}_{n}^{}Σ_{n} Spin_{n}. Here Spin_{n} acts from the right on { \atop GL}_{n}^{} through the inclusion and from the left on Σ_{n} through the spin representation. Note that ΣM is a vector bundle over _{}^{2}T^{}M, but not over M.

A section Φ Γ ΣM determines a Riemannian metric g g_{Φ} and a g-spinor φ φ_{Φ} Γ Σ_{g}M and vice versa. Therefore we henceforth identify Φ with g, φ . Next, we denote by S ΣM the universal bundle of unit spinors, i.e.

S ΣM Φ ΣM Φ 1

where Φ φ_{Φ} _{g_{Φ}}. Finally we introduce the spaces of smooth sections

Γ ΣM and Γ S ΣM

which we also regard as Fréchet fiber bundles over . Here and elsewhere in the article Γ ΣM and Γ S ΣM denote the spaces of sections of the corresponding bundles over M, and not over _{}^{2}T^{}M.

3.2. The Bourguignon-Gauduchon horizontal distribution. In order to compare different fibers Σ_{g_{0}}M and Σ_{g_{1}}M of the universal spinor bundle over some point x M, we shall need the natural horizontal distribution of Bourguignon and Gauduchon. We refer to [9] for details. Let V be an oriented real vector space. We denote by V the set of oriented bases of V which we think of as the set of orientationpreserving linear isomorphisms ℝ^{n} V. By the polar decomposition theorem V is diffeomorphic to the product of the set of special orthogonal matrices and the cone of positive-definite symmetric matrices. Hence the natural fibration p V _{}^{2}V where the fiber _{g}V over g consists precisely of the set of oriented g-orthonormal bases, is trivial. Using b V to identify V with ℝ^{n}, the tangent space of V at b can be decomposed into the space tangent to the fibre T_{b}^{v} Λ^{2}ℝ^{n}, and the space of symmetric matrices T_{b}^{h} ^{2}ℝ^{n} which is horizontal. Since the distribution b T_{b}^{v} is SO_{n}-equivariant we get a connection dubbed natural by Bourguignon and Gauduchon. This construction can be immediately generalised to the universal cover ˜ V _{}^{2}V, where the fiber _{ g̃} V is diffeomorphic to Spin_{n}.

By naturality of the pointwise construction above we obtain a horizontal distribution of the principal Spin_{n}-bundle P̃ _{}^{2}M and consequently one of ΣM: The fiber over g_{x} _{}^{2}M is given by T_{x}M. For every b̃ P̃_{g_{x}} lying over the basis b of T_{x}M, we have a complement to the tangent space of the fiber (isomorphic to the skew-adjoint endomorphisms of T_{x}M) given by the symmetric endomorphisms of T_{x}M. Using the projection we can identify this space with ^{2}T_{x}^{}M, the tangent space of _{}^{2}M at x.

3.3. Parallel transport. The horizontal distribution just defined gives rise to a “partial” connection on ΣM _{}^{2}M in the sense that _{X}ψ is defined for a smooth spinor ψ if X is vertical for the bundle map _{}^{2}M M. This allows us to compare the fibers of Σ_{g_{0}}M and Σ_{g_{1}}M over a given point x M along a path from g_{1} x to g_{2} x , but not over two different points. There are two ways of making this comparison.

Following again [9], we consider for two given metrics g_{0} and g_{1} over T_{x}M the endomorphism A_{g_{0}}^{g_{1}} defined by g_{1} v, w g_{0} A_{g_{0}}^{g_{1}}v, w . The endomorphism A_{g_{0}}^{g_{1}} is self-adjoint and positive definite with respect to g_{0}. Consequently, it has a positive self-adjoint square root B_{g_{0}}^{g_{1}}. One easily checks that A_{g_{1}}^{g_{0}} is the inverse of A_{g_{0}}^{g_{1}}, whence B_{g_{0}}^{g_{1}} ^{1} B_{g_{1}}^{g_{0}}. We map a g_{0}-orthonormal basis b to B_{g_{1}}^{g_{0}} b A_{g_{1}}^{g_{0}} ^{1 2}b P_{g_{1}}. This

map can be lifted to a map B̃_{g_{1}}^{g_{0}}  P̃_{g_{0}}  P̃_{g_{1}} by sending the spinor b̃, φ̃ to B̃_{g_{1}}^{g_{0}}b̃, φ̃
which induces an isometry B̂_{g_{1}}^{g_{0}}  _{g_{0}}  _{g_{1}}. In particular, we obtain a map B̂_{g_{1}}^{g_{0}}
g_{0}  _{g_{1}}. This isometry coincides with the parallel transport P̃_{g_{0}}  P̃_{g_{1}} along

the path g_{t} _{}^{2}T_{x}^{}M associated with the Bourguignon–Gauduchon distribution if g_{t} 1 t g_{0} tg_{1} is just linear interpolation [9, Proposition 2].

An alternative description of the resulting parallel transport can be also given in terms of the generalised cylinder construction from [6]. This works for an arbitrary piecewise smooth path g_{t} I 0, 1 from g_{0} to g_{1}. Let C I M be the Riemannian product with metric G dt^{2} g_{t}. The spin structure on M and the unique spin structure on 0, 1 induce a product spin structure on the cylinder. Thus we obtain a spinor bundle Σ_{g_{t}}M on M for t 0, 1 and a spinor bundle Σ_{G}C on the cylinder. Each such spinor bundle carries a connection denoted by g_{t} resp. ^{C} coming from the Levi-Civita connection, and a Clifford multiplication T M Σ_{g_{t}}M Σ_{g_{t}}M, X φ X _{t} φ, resp. T M Σ_{G}C Σ_{G}C, X φ X φ. To lighten notation we simply write X φ, though Clifford multiplication actually depends on t. Note that our notation differs from the one in [6]:

Notation in [6] X φ X _{t} φ ; Present article X φ X φ

We also write M_{t} for the Riemannian manifold t M, g_{t} and ν ∂_{t} for the canonical vector field on C which is normal to M_{t}. If n is even, then ΣM_{t} ΣC _{M_{t}}, but Clifford multiplication is not preserved by restriction. Indeed, we have

X φ ν X φ  (4)

for X Γ T M and φ Γ ΣM_{t} . The same holds for n odd if we set ΣM Σ^{}C. Parallel transport on C gives rise to linear isometries B̃_{g_{t}}^{g} Σ_{g,x}M Σ_{g_{t},x}M_{t} along the curves t t, x which coincide indeed with the maps B̃_{g_{t}}^{g} defined above if we use linear interpolation g_{t} tg_{1} 1 t g_{0} (cf. Section 5 in [6]). In the sequel we will therefore think of the distribution at Φ g, φ as

Φ \frac{d}{dt} _{t 0} B̂_{g_{t}}^{g}φ g_{t} a path with g_{0} g π φ ,

which we can identify with the space of symmetric 2-forms ^{2}T^{}M. Consequently we obtain the decomposition

T_{g,φ}ΣM ^{2}T_{x}^{}M Σ_{g,x}M,  (5)

where g, φ ΣM has basepoint x M. Passing to the Fréchet bundle , we obtain in view of the generalised cylinder construction a horizontal distribution also denoted by . At Φ g, φ it is given by

Φ \frac{d}{dt} _{t 0} B̃_{g_{t}}^{g}φ g_{t} a path with g_{0} g π φ .

It follows that

T_{Φ} _{Φ} T_{Φ} _{g} Γ ^{2}T^{}M Γ Σ_{g}M ,

and the subspace T_{Φ} corresponds to

Γ ^{2}T^{}M Γ φ_{Φ}^{}h, ψ Γ ^{2}T^{}M Γ Σ_{g}M ψ x , φ_{Φ} x 0 x M .

4. The spinorial energy functional

In this section we introduce the spinorial energy functional and investigate its basic properties.

Definition 4.1. The energy functional is defined by

ℝ_{0}, Φ \frac{1}{2} _{M} ^{g}φ _{g}^{2} dv^{g},

where, as above, we identify an element Φ with the pair g, φ g_{Φ}, φ_{Φ} for g Γ _{}^{2}T^{}M and φ Γ Σ_{g}M .

4.1. Symmetries of the functional.

4.1.1. Rescaling. Consider the action of c ℝ_{}on the cone of metrics by rescaling g c^{2}g. This conformal change of the metric can be canonically lifted to the spinor bundle. In the notation of Subsection 3.2 we have A_{g}^{c^{2}g} c^{2}Id, B_{g}^{c^{2}g} cId, B_{c^{2}g}^{g} c^{1}Id. As before we obtain maps B̃_{c^{2}g}^{g} P̃_{g} P̃_{c^{2}g} and a fiberwise isometry B̂_{c^{2}g}^{g} Σ_{g} Σ_{c^{2}g} sending b̃, φ̃ to B̃_{c^{2}g}^{g} b̃ , φ̃ . In particular, we get the bundle map

B̂_{c^{2}g}^{g} , φ _{g} B̂_{c^{2}g}^{g}φ _{c^{2}g}.

Proposition 4.2. Let Φ g, φ . Then c^{2}g, B̂_{c^{2}g}^{g}φ c^{n 2} g, φ for all c 0.

Proof. If g̃ c^{2}g for some constant c 0, then g̃ ^{g} (cf. for instance Lemma II.5.27 in [31]). Further, dvg̃ c^{n} dv^{g} and ẽ_{k} c^{1}e_{k}, so that

c^{2}g, B̂_{c^{2}g}^{g}φ \frac{1}{2} _{M} _{k 1}^{n} g̃ ; ẽ_{k} B̂_{c^{2}g}^{g}φ ^{2} dvg̃ \frac{1}{2} _{M} _{k 1}^{n} _{c^{1} e_{k}}^{g}φ ^{2}c^{n} dv^{g} c^{n 2} g, φ

which is what we wanted to show. □

4.1.2. Action of the spin-diffeomorphism group. An orientation preserving diffeomorphism f M M induces a bundle map df P P, where df maps an oriented frame v_{1}, …, v_{n} over x to the oriented frame df v_{1} , …, df v_{n} over f x . Then f is called spin structure preserving if it lifts to a bundle map F P̃ P̃ of the spin structure P̃ P making the diagram

\begin{matrix}P̃ & ^{F} & P̃ \\  ;  ;  ;  & &  ;  ;  ;  \\ P & ^{df} & P \end{matrix}  (6)

commutative. In other words, an orientation preserving diffeomorphism f M M is spin structure preserving if and only if the pullback of P̃ P under f is a spin structure on M that is equivalent to P̃ P. We denote by Diff_{s} M Diff_{}M the group of spin structure preserving diffeomorphisms. A spin-diffeomorphism is a diffeomorphism F P̃ P̃ making the diagram (6) commutative for some orientation preserving diffeomorphism f M M. Since P̃ P is a ℤ_{2}-principal bundle the lift is determined up to a ℤ_{2}-action. Put differently the group of all spin-diffeomorphisms which we denote by { \atop Diff}_{s} M , is an extension of Diff_{s} M by ℤ_{2} 1 , i.e. it fits into a short exact sequence

1 1 { \atop Diff}_{s} M Diff_{s} M 1.

A special situation arises if f Diff_{0} M , i.e. f is isotopic to the identity. Since the homotopy can be lifted to P̃ and homotopic isotopies yield the same lift, we obtain a map { \atop Diff}_{0} M { \atop Diff}_{s} M from the universal covering group of Diff_{0} M .

If g is a Riemannian metric on M, then F { \atop Diff}_{s} M covering f Diff_{s} M maps P̃_{g} to P̃_{f_{}g}, where by definition f_{}g f^{1} ^{}g. For any φ _{g} we obtain a spinor field F_{}φ _{f_{}g} as follows: If φ is locally expressed as b̃, φ̃ , then F_{}φ is locally expressed as F b̃ f^{1}, φ̃ f^{1} . Since

F_{}φ _{f_{}g} x φ̃ f^{1} x , φ̃ f^{1} x  φ _{g} f^{1} x ,  (7)

F_{}is a map from _{g} to _{f_{}g} with inverse F^{1} _{}. Consequently, we obtain the bundle map

F_{}, φ _{g} F_{}φ _{f_{}g}.

Note that F_{1} F_{2} _{}F_{1} F_{2}. Since the spinorial energy functional only depends on the metric and the spinor bundle which both transform naturally under spindiffeomorphisms, we immediately conclude the following

Proposition 4.3. Let Φ g, φ . Then F_{}Φ Φ for all F { \atop Diff}_{s} M .

Finally we discuss the infinitesimal action of { \atop Diff}_{s} M on . Consider a vector field X Γ T M with associated flow f_{t} Diff_{0} M , that is \frac{d}{dt}f_{t} X f_{t} with f_{0} id_{M}. Hence f_{t} lifts to a 1-parameter family F_{t} { \atop Diff}_{s} M with F_{0} idP̃. If F { \atop Diff}_{s} M we define F^{}by F^{}F_{}^{1}, so that F^{}φ _{f^{}g} if F covers f Diff M . Hence F_{t}^{}Φ is a family of sections of ΣM which we can differentiate with respect to t. Using the connection on the bundle we may split the resulting “Lie derivative”

\frac{d}{dt} _{t 0} F_{t}^{}Φ T_{Φ} _{Φ} T_{Φ} _{g}

into the horizontal part

Xg \frac{d}{dt} _{t 0} f_{t}^{}g Γ ^{2}T^{}M _{Φ}

and the vertical part

˜_{X}^{g} φ \frac{d}{dt} _{t 0} B̂_{g}^{f_{t}^{}g}F_{t}^{}Φ Γ Σ_{g}M T_{Φ} _{g}.

This vertical part ˜ _{X}^{g} φ is called the metric Lie derivative. It is the lift from the metric Lie derivative ^{g} on tensor fields which is defined using the metric, and satisfies, in particular, _{X}^{g}g 0 (cf. [9]). By [9, Proposition 17] we have

˜_{X}^{g} φ _{X}^{g}φ \frac{1}{4}dX^{}φ.  (8)

More generally, consider a curve Φ_{t} g_{t}, φ_{t} in with ġ_{t}, φ̇_{t} \frac{d}{dt}Φ_{t} T_{Φ_{t}} and X_{t} a time-dependent vector field whose flow lifts to F_{t} { \atop Diff}_{s} M . Since F^{} is linear on the fibres as a map , we get

\frac{d}{dt}F_{t}^{}Φ_{t} F_{t}^{}_{X_{t}}g_{t} ġ_{t}, ˜ _{X_{t}}^{g} φ_{t} φ̇_{t} .  (9)

4.1.3. Symmetries from pointwise representation theory. Let P_{G} M be a principal G-fibre bundle with connection, V a G-representation space and L End V a G-equivariant map. Then it follows from general principal fibre bundle theory that L gives rise to a well-defined bundle endomorphism of P_{G} _{G} V which is parallel with respect to the induced covariant derivative . Using the representation theory of spinors we will compute the G Spin n -equivariant maps for the spin representations. Those which in addition preserve the inner products will therefore preserve the energy functional, i.e. g, L φ g, φ , cf. Tables 1 and 2 below. For example, the complex volume form vol^{ℂ} i^{n n 1 2} vol_{g} acts as an isometry on Σ_{n} via Clifford multiplication and commutes with Spin_{n}. Further, if n is even, it defines an involution so that not only vol^{ℂ} is preserved under ^{g}, but also the decomposition Σ_{n} Σ_{n}^{}Σ_{n}^{}into the -eigenspaces of positive and negative eigenspinors (these are actually the irreducible Spin_{n}-representations mentioned in Section 2). Consequently, a positive or negative spinor is a critical point of if and only if it is a critical point of the restriction of to positive and negative spinors. A further application will be discussed in Section 6.2. To lighten notation we shall drop any reference to background metrics.

Since we work with a real inner product on spinors it will be convenient to work with ℝ-linear maps. Though we mainly consider complex spin representations we start by considering the real spin representations Σ_{n}^{ℝ}. By definition, these are obtained by restricting the irreducible real representations of Cliff_{n} to Spin_{n} (cf. [31, Definition I.5.11]). The algebra End Σ_{n}^{ℝ} ^{Spin^{n}} of Spin_{n}-equivariant endomorphisms of Σ_{n}^{ℝ} can be computed from the decomposition of Σ_{n}^{ℝ} into irreducible components and their types. Recall that by Schur’s lemma for an irreducible real representation V of Spin_{n} the algebra End V ^{Spin^{n}} is either ℝ, ℂ or ℍ which by definition is the type of the representation. Note that Σ_{n}^{ℝ} is not always irreducible as a representation of Spin_{n}, see [31, Proposition I.5.12] for the precise decomposition. However, the irreducible components always have the same type, which we will refer to as the type of Σ_{n}^{ℝ}. As Spin_{n} generates Cliff_{n}^{ev} as an algebra, an endomorphism is Cliff_{n}^{ev} equivariant if and only if it is Spin_{n} equivariant. Thus the type of Σ_{n}^{ℝ} is determined by the representation of the even part of Cliff_{n} which as an algebra is just Cliff_{n 1}. Hence the type of Σ_{n}^{ℝ} can be read off from Table III in [31, Section I.5] (cf. also [31, Remark I.5.13]). With this data at hand, End Σ_{n}^{ℝ} ^{Spin^{n}} and the group Isom Σ_{n}^{ℝ} ^{Spin^{n}} of Spin_{n}-equivariant isometries of Σ_{n}^{ℝ} can easily be determined, see Table 1. Here, 𝕂 i denotes the i i-matrices with coefficients

n mod 8 type of Σ_{n}^{ℝ} decomposition End Σ_{n}^{ℝ} ^{Spin^{n}} Isom Σ_{n}^{ℝ} ^{Spin^{n}}
0  ℝ  Δ_{}Δ_{}ℝ 1 ℝ 1  O 1 O 1
1  ℝ  Δ Δ  ℝ 2  O 2
2  ℂ  Δ Δ  ℂ 2  U 2
3  ℍ  Δ  ℍ 1  Sp 1
4  ℍ  Δ_{}Δ_{}ℍ 1 ℍ 1  Sp 1 Sp 1
5  ℍ  Δ  ℍ 1  Sp 1
6  ℂ  Δ  ℂ 1  U 1
7  ℝ  Δ  ℝ 1  O 1
Table 1. Isomorphism classes of Isom Σ_{n}^{ℝ} ^{Spin^{n}}

in 𝕂 ℝ, ℂ or ℍ. Further, Δ Δ means that Σ_{n}^{ℝ} is the sum of two equivalent irreducible Spin_{n}-representations, while Δ_{}Δ_{}is an irreducible decomposition into non-equivalent ones. For the remaining cases Σ_{n}^{ℝ} is irreducible. The algebra End_{ℝ} rΣ_{n} ^{Spin^{n}} of ℝ-linear Spin_{n}-equivariant endomorphisms of Σ_{n} and the group of ℝ-linear Spin_{n}-equivariant isometries Isom_{ℝ} Σ_{n} ^{Spin^{n}} can now be computed as follows. The complex representation Σ_{n} is obtained as the complexification of Σ_{n}^{ℝ} if n 0, 6 or 7 mod 8 so that as a real representation Σ_{n} Σ_{n}^{ℝ} Σ_{n}^{ℝ}. In the remaining cases Σ_{n} Σ_{n}^{ℝ}, and therefore End_{ℝ} Σ_{n} ^{Spin^{n}} End Σ_{n}^{ℝ} ^{Spin^{n}}. Now if for instance n 6 mod 8, then Σ_{n}^{ℝ} Δ is irreducible and End Σ_{n}^{ℝ} ^{Spin^{n}} ℂ 1 according to Table 1. Hence Σ_{n} Δ Δ as a real representation and End_{ℝ} Σ_{n} ^{Spin^{n}} ℂ 2 as a real algebra. Continuing in this vein we arrive at Table 2.

For later applications we consider some concrete elements in Isom_{ℝ} Σ_{n} ^{Spin^{n}}. Obvious ones are scalar multiplication by id Isom_{ℝ} Σ_{n} ^{Spin^{n}} and S^{1} Isom_{ℝ} Σ_{n} ^{Spin^{n}} as well as the action by the volume element vol_{g}. Further elements of Isom_{ℝ} Σ_{n} ^{Spin^{n}} are provided by quaternionic and real structures, i.e. complex anti-linear maps J with J^{2} id and J^{2} id respectively. For n 0, 1, 6 or 7 mod 8 there exists a Spin_{n}-equivariant quaternionic structure J_{n} on Σ_{n} while there exists a Spin_{n}- equivariant real structure J_{n} for n 2, 3, 4 or 5 mod 8 [19, Sec. 1.7]. For n is even we obtain further real and quaternionic structures J̃_{n} J_{n} vol . Isom_{ℝ} Σ_{n} ^{Spin^{n}}, namely J̃_{n}^{2} 1 for n 0, 2 mod 8 and J̃_{n}^{2} 1 for n 4, 6 mod 8 [25, Chapter 2].

Corollary 4.4. On Σ_{n} there are real structures in dimensions n 0, 1, 2, 6, 7 mod 8 and quaternionic structures in dimensions n 2, 3, 4, 5, 6 mod 8 preserving .

n mod 8 End_{ℝ} Σ_{n} ^{Spin^{n}} Isom_{ℝ} Σ_{n} ^{Spin^{n}}
0  ℝ 2 ℝ 2  O 2 O 2
1  ℝ 2  O 2
2  ℂ 2  U 2
3  ℍ 1  Sp 1
4  ℍ 1 ℍ 1  Sp 1 Sp 1
5  ℍ 1  Sp 1
6  ℂ 2  U 2
7  ℝ 2  O 2
Table 2. Isomorphism classes of Isom_{ℝ} Σ_{n} ^{Spin^{n}}

Together with scalar multiplication by S^{1} each real structure yields a subgroup of Isom_{ℝ} Σ_{n} ^{Spin^{n}} isomorphic to S^{1} ℤ_{2} O 2 , where ℤ_{2} acts by conjugation on S^{1}. On the other hand, a quaternionic structure turns Σ_{n} into a quaternionic vector space so we get an induced equivariant and isometric action of Sp 1 . This actually gives the whole group Isom_{ℝ} Σ_{n} ^{Spin^{n}} for n odd. If n 2 mod 8, then the action of J_{n} and i generates a subgroup isomorphic to SU 2 . Further, vol_{g} induces a complex linear map with vol_{g}^{2} id and which commutes with J_{n} and i. One easily checks that T_{α} cos α id sin α vol Isom rΣ_{n} ^{Spin^{n}}. As T_{π} coincides with id SU 2 we obtain the group S^{1} _{ℤ_{2}} SU 2 U 2 . For n 6 mod 8 the symmetries J̃_{n} and i give rise to an SU 2 -action on the unit spinors. The group generated by J_{n} and this SU 2 is a semi-direct product rather than a direct product of ℤ_{2} with SU 2 , for J_{n} and i anti-commute. For n 4 mod 8 we get a proper subgroup of Isom_{ℝ} Σ_{n} ^{Spin^{n}} isomorphic to ℤ_{2} SU 2 while for n 0 mod 8 we get a proper subgroup isomorphic to ℤ_{2} O 2 .

4.2. Critical points. Next we determine the critical points and compute the L^{2}- gradient of .

Let Φ g, φ . In view of the splitting (5), any element in T_{Φ} can be decomposed into a vertical part φ̇ \frac{d}{dt} _{t 0} φ_{t} for a curve φ_{t} in _{g} with φ_{0} φ, and a horizontal part ġ^{hor} \frac{d}{dt} _{t 0} g_{t}^{hor}, where g_{t}^{hor} is the horizontal lift of a curve g_{t} in with g_{0} g starting at Φ. In the following we identify ġ^{hor} T_{Φ} with ġ \frac{d}{dt} _{t 0} g_{t} T_{g} Γ ^{2}T^{}M without further notice. The pointwise constraint φ_{t} 1 implies that φ̇ must be pointwise perpendicular to φ, i.e. φ̇ Γ φ^{}. We define Q_{1} Φ Γ ^{2}T^{}M via

Q_{1} Φ , ġ _{g}  \frac{d}{dt} _{t 0} g_{t}^{hor}  D_{Φ} ġ^{hor}, 0 .

Up to the minus sign this is just the energy-momentum tensor for φ as defined in [6]. Similarly, we introduce

Q_{2} Φ , φ̇ _{g}  \frac{d}{dt} _{t 0} g, φ_{t}  D_{Φ} 0, φ̇

where φ, ψ _{g} _{M} φ, ψ dv^{g} denotes the natural L^{2}-inner product on _{g}. Then under the identification of (5),

Q  T , Φ  Q_{1} Φ , Q_{2} Φ Γ ^{2}T^{}M Γ φ^{}

becomes the negative gradient of .

The compatibility of with the actions of ℝ_{}and { \atop Diff}_{s} M implies the following equivariance properties of Q. In particular, we obtain

Corollary 4.5. Let Φ g, φ . Then

(i) Q_{1} c^{2}g, B̂_{c^{2}g}^{g}φ Q_{1} g, φ and Q_{2} c^{2}g, B̂_{c^{2}g}^{g}φ c^{2}Q_{2} g, φ for all c 0.

(ii) Q F_{}Φ F_{}Q Φ for all F { \atop Diff}_{s} M .

Proof. (i) By definition of the negative gradient one has

Q_{1} c^{2}g, B̂_{c^{2}g}^{g}φ , c^{2}ġ _{c^{2}g}  D_{c^{2}g,B_{c^{2}g}^{g}φ} c^{2}ġ, 0
\frac{d}{dt} _{t 0} c^{2} g tġ , B_{c^{2} g tġ}^{g} φ
c^{n 2} D_{g,φ} ġ, 0 c^{n 2} Q_{1} g, φ , ġ _{g},

where for the last line we have used Proposition 4.2. On the other hand,

Q_{1} c^{2}g, B̂_{c^{2}g}^{g}φ , c^{2}ġ _{c^{2}g} c^{n 2} Q_{1} c^{2}g, B̂_{c^{2}g}^{g}φ , ġ _{g},

whence the result for Q_{1}. Similarly,

c^{n} Q_{2} c^{2}g, B̂_{c^{2}g}^{g}φ , φ̇ _{g} Q_{2} c^{2}g, B̂_{c^{2}g}^{g}φ , φ̇ _{c^{2}g} c^{n 2} Q_{2} g, φ , φ̇ _{g}.

(ii) Follows from Proposition 4.3. □

The formal adjoint δ_{g}^{}of the divergence δ_{g} Γ ^{2}T^{}M Γ T^{}M , is the symmetrisation of the covariant derivative, i.e. δ_{g}^{}η X, Y \frac{1}{2} _{X}^{g}η Y _{Y}^{g} η X for η Γ T^{}M . In particular, _{X}g 2δ_{g}^{}X^{}. Since δ_{g}^{}is overdetermined elliptic we have an orthogonal decomposition Γ ^{2}T^{}M ker δ_{g} im δ_{g}^{}(see e.g. Appendix I, Corollary 32 in [8]). If we identify T_{g} Γ ^{2}T^{}M , then im δ_{g}^{}is tangent to the Diff M -orbit through g and ker δ_{g} provides a natural complement in T_{g} . In a similar fashion we consider the operator

λ_{g,φ}^{}Γ T M T_{g,φ} Γ ^{2}T^{}M Γ Σ_{g}M
X _{X}g, ˜ _{X}^{g} φ 2δ_{g}^{}X^{}, _{X}^{g}φ \frac{1}{4}dX^{}φ .

Consider the fiberwise linear maps A_{φ} T M Σ_{g}M, X _{X}φ and B_{φ} Λ^{2}T^{}M Σ_{g}M, ω ω φ. Expressed in a local orthonormal frame e_{i} , their pointwise adjoints A_{φ}^{}and B_{φ}^{}are given by

A_{φ}^{}φ̇ _{i} φ̇, _{e_{i}}^{g}φ e_{i} and B_{φ}^{}φ̇ _{i ȷ} φ̇, e_{i} e_{j} φ e_{i} e_{j}.

Hence λ_{g,φ}^{}X 2δ_{g}^{}X^{}, A_{φ} X \frac{1}{4}B_{φ} dX^{}and further

λ_{g,φ} Γ ^{2}T^{}M Γ Σ_{g}M Γ T M
ġ, φ̇ 2 δ_{g}ġ ^{}A_{φ}^{}φ̇ \frac{1}{4} δ_{g}B_{φ}^{}φ̇ ^{}

for the formal adjoint of λ_{g,φ}^{}. As in the case of metrics, im λ_{g,φ}^{}is tangent to the { \atop Diff}_{s} M -orbit through g, φ . Note that im λ_{g,φ}^{}is actually contained in T_{g,φ} Γ ^{2}T^{}M Γ φ^{}, since the { \atop Diff}_{s} M -action preserves .

Lemma 4.6. There are orthogonal decompositions:

(i) T_{g,φ} ker λ_{g,φ} im λ_{g,φ}^{}for g, φ . (ii) T_{g,φ} ker λ_{g,φ} T_{g,φ} im λ_{g,φ}^{}for g, φ .

Proof. (i) The symbol of λ_{g,φ}^{}at ξ T^{}M_{x} is given by

σ_{ξ} λ_{g,φ}^{}v i ξ v^{}v^{}ξ, \frac{1}{4}ξ v^{}φ

for v T_{x}M and is clearly injective. Hence λ_{g,φ}^{}is overdetermined elliptic and the result follows as above. (ii) If φ _{g} is a unit spinor, then _{ X̃} φ Γ φ^{}for any X Γ T M . Hence the image of λ_{g,φ}^{}is contained in T_{g,φ} and the result follows from (i). □

Just as the classical Bianchi identity of the Riemann curvature tensor is a consequence of its Diff M -equivariance [28], we note the following corollary to Proposition 4.3.

Corollary 4.7 (Bianchi identity). Let g, φ . Then λ_{g,φ}Q g, φ 0.

Proof. Since is invariant under { \atop Diff}_{s} M according to Proposition 4.3, the orbit { \atop Diff}_{s} M g, φ is contained in the level set ^{1} g, φ . Now the (negative) gradient of is orthogonal to the level set, hence in particular to { \atop Diff}_{s} M g, φ . At g, φ , the tangent space to this orbit is given by im λ_{g,φ}^{}. The result follows in view of Lemma 4.6. □

We recall the definition of the connection Laplacian ^{g} ^{g}  _{g}  _{g}, namely

g ^{g}φ _{j 1}^{n} _{e_{j}}^{g} _{e_{j}}^{g}φ _{_{e_{j}}^{g} e_{j}}^{g} φ ,

where e_{1}, …, e_{n} is a local orthonormal basis. Note that ^{g}φ, ^{g}φ _{g} ^{g} ^{g}φ, φ _{g}. For a tensor T let

div_{g} T _{k 1}^{n} _{e_{k}}^{g}T e_{k},

be the divergence of T.

Theorem 4.8. For Φ g, φ we have

\begin{matrix}Q_{1} Φ \frac{1}{4} ^{g}φ _{g}^{2}g \frac{1}{4} div_{g} T_{g,φ} \frac{1}{2} ^{g}φ ^{g}φ , \\ Q_{2} Φ ^{g} ^{g}φ ^{g}φ _{g}^{2}φ, \end{matrix}

where T_{g,φ} Γ T^{}M ^{2}T^{}M is the symmetrisation (as defined in Section 2) in the second and third component of the 3, 0 -tensor defined by X Y φ, _{Z}^{g}φ for X, Y and Z in Γ T M . Further, ^{g}φ ^{g}φ is the symmetric 2-tensor defined by ^{g}φ ^{g}φ X, Y _{X}^{g}φ, _{Y}^{g} φ for X, Y Γ T M .

Remark 4.9. Let D_{g}  _{g}  _{g} be the Dirac operator associated with the spin

structure, i.e. locally D_{g}φ _{k} e_{k} _{e_{k}}^{g}φ. A pointwise computation with ^{g}e_{k} x 0 implies that Tr_{g} div_{g} T_{g,φ} D_{g}^{2}φ, φ D_{g}φ ^{2}, that is, the trace of the divergence term in Q_{1} measures the pointwise failure of self-adjointness of D_{g}.

Proof. The vertical variation of which gives Q_{2} is the easy part. For g a fixed Riemannian metric let φ_{t} _{g} be a smooth family of spinors with φ_{0} φ. Set φ̇ \frac{d}{dt} _{t 0} φ_{t}. Then

\frac{d}{dt} _{t 0} g, φ_{t}  \frac{1}{2} _{M} \frac{d}{dt} _{t 0} ^{g}φ_{t} _{g}^{2} dv^{g}  _{M} ^{g}φ̇, ^{g}φ _{g} dv^{g}.

To get Q_{2} Φ it remains to determine the component of ^{g} ^{g}φ orthogonal to φ. Since φ, _{X}^{g}φ 0 for all X Γ T M , we have

φ, ^{g} ^{g}φ _{k} φ, _{e_{k}}^{g} _{e_{k}}^{g}φ ^{g}φ _{g}^{2},

and we get the asserted formula for Q_{2}. Secondly, we calculate Q_{1}. Let g_{t} be a smooth family of Riemannian metrics with g_{0} g. Set ġ \frac{d}{dt} _{t 0} g_{t}. Further, let g_{t}^{hor} g_{t}, φ_{t} be the horizontal lift of the family g_{t} to . Here and in the following we use the notation of Section 3.3. Then

\frac{d}{dt} _{t 0} g_{t}^{hor}  \frac{1}{4} _{M} ^{g}φ _{g}^{2} Tr_{g}ġ dv^{g} \frac{1}{2} _{M} \frac{d}{dt} _{t 0} ^{g_{t}}φ_{t} _{g_{t}}^{2} dv^{g},  (10)

where we have used the standard variation formula \frac{d}{dt} _{t 0} dv^{g_{t}} \frac{1}{2}Tr_{g}ġ dv^{g}. For the second term we first proceed pointwise and fix a local orthonormal basis e_{k,t} in C around x 0 M with e_{0,t} ν. For e_{k,0} we simply write e_{k}. We may assume that ge_{k} x 0 and _{ν}^{C}e_{k,t} 0 for k 1, …, n. It follows that

g_{t}φ_{t} _{g_{t}}^{2} _{k 1}^{n} _{e_{k,t}}^{g_{t}}φ_{t} ^{2}

and thus

\frac{d}{dt} _{t 0} ^{g_{t}}φ_{t} _{g_{t}}^{2} 2 _{k 1}^{n} _{ν}^{C} _{e_{k,t}}^{g_{t}}φ_{t}, _{e_{k}}^{g}φ _{t 0}

(recall that ν ∂_{t}). On the other hand let W_{t} denote the Weingarten map T M_{t} T M_{t} for the hypersurface M_{t} t M defined through

_{X}^{C}Y _{X}^{g_{t}}Y g_{t} W_{t} X , Y ν.

We identify 0 M with M and simply write W for W_{0} in the following. By (3.5) in [6] and equation (4) we have

_{X}^{g_{t}}φ_{t}  _{X}^{C}φ_{t} \frac{1}{2}ν W_{t} X φ_{t}  _{X}^{C}φ_{t} \frac{1}{2}W_{t} X φ_{t}.  (11)

For simplicity we assume from now on that _{ν}^{C}X 0. Note that this implies in particular that _{ν}^{C} W_{t} X _{ν}^{C}W_{t} X . Since _{ν}^{C}φ_{t} 0 we get

_{ν}^{C} _{X}^{g_{t}}φ_{t} _{t 0}  _{ν}^{C} _{X}^{C}φ_{t} \frac{1}{2}W_{t} X φ_{t}  _{t 0}
R^{C} ν, X φ _{ν,X}^{C}φ \frac{1}{2} _{ν}^{C}W X φ,

where R^{C} is the curvature operator acting on Γ ΣC , cf. equation (3). We investigate the first term of the bottom line. With e_{0} ν we obtain

R^{C} ν, X φ \frac{1}{2} _{0 i j n} R^{C} ν, X e_{i}, e_{j} _{g} e_{i} e_{j} φ

and further

\frac{1}{2} _{0 i j n} R^{C} ν, X e_{i}, e_{j} _{g} e_{i} e_{j} φ

\frac{1}{2} _{1 i j n} R^{C} e_{i}, e_{j} X, ν _{g} e_{i} e_{j} φ \frac{1}{2} _{j 1}^{n} R^{C} X, ν ν, e_{j} _{g} ν e_{j} φ

\frac{1}{2} _{1 i j n} R^{C} e_{i}, e_{j} X, ν _{g} e_{i} e_{j} φ \frac{1}{2} _{j 1}^{n} R^{C} X, ν ν, e_{j} _{g} e_{j} φ

\frac{1}{2} _{1 i j n} R^{C} e_{i}, e_{j} X, ν _{g} e_{i} e_{j} φ \frac{1}{2} _{j 1}^{n} W^{2} X  _{ν}^{C}W X , e_{j} _{g} e_{j} φ
\frac{1}{2} _{1 i j n} R^{C} e_{i}, e_{j} X, ν _{g} e_{i} e_{j} φ \frac{1}{2}W^{2} X φ \frac{1}{2}  _{ν}^{C}W X φ,

where we have used the Riccati equation _{ν}^{C}W_{t} X R^{C} X, ν ν W_{t}^{2} X . For the second term we observe that ν, X _{X}^{C}ν W X . Using the relation _{W X}^{C}φ _{W X}^{g}φ \frac{1}{2}W^{2} X φ, which follows from equation (11), and (4.3) from [6] we finally obtain

_{ν}^{C} _{X}^{g_{t}}φ_{t} _{t 0}  _{W X}^{g}φ \frac{1}{4} _{1 i j n}  _{e_{i}}^{g} ġ e_{j}, X  _{e_{j}}^{g} ġ e_{i}, X e_{i} e_{j} φ. (12)

Since W X ġ X, 2 by (4.1) in [6], upon substituting X e_{k} the first term of (12) contributes

\frac{1}{2} _{M} _{k 1}^{n} ġ e_{k}, e_{j}  _{e_{j}}^{g}φ, _{e_{k}}^{g}φ dv^{g}  \frac{1}{2} _{M} ġ, ^{g}φ ^{g}φ _{g} dv^{g}

to the integral in (10) (with , _{g} denoting the pointwise inner product on ^{2}T^{}M induced by g). Further, contracting the second term of (12) with _{e_{k}}^{g}φ for X e_{k}, and taking the sum over k gives

\frac{1}{4} _{k 1}^{n} _{i,j 1,i j}^{n} _{e_{i}}^{g} ġ e_{j}, e_{k} e_{i} e_{j} φ, _{e_{k}}^{g}φ

\frac{1}{4} _{i,j,k 1}^{n} _{e_{i}}^{g} ġ e_{j}, e_{k} e_{i} e_{j} φ, _{e_{k}}^{g}φ
\frac{1}{4} ^{g}ġ, T_{g,φ} _{g}.

Finally, a pointwise computation using a local basis with _{e_{j}}^{g}e_{k} 0 in the given point yields

gġ, T_{g,φ} _{g} _{k 1}^{n} _{e_{k}}^{g} ġ, T_{g,φ} e_{k}, , _{g}

_{k 1}^{n} e_{k} ġ, T_{g,φ} e_{k}, , _{g} _{k 1}^{n} ġ, _{e_{k}}^{g}T_{g,φ} e_{k}, , _{g}.

The first sum vanishes is the divergence of a vector field, and thus vanishes after integration. The second sum is just the pointwise inner product of ġ with the divergence of T_{g,φ}. We obtain

\frac{1}{4} _{M} ^{g}ġ, T_{g,φ} _{g} dv^{g} \frac{1}{4} _{M} ġ, div T_{g,φ} _{g} dv^{g},

and the assertion follows. □

In the following we determine the critical points of , i.e. pairs g, φ satisfying the system of Euler–Lagrange equations

\begin{matrix}^{g}φ _{g}^{2}g div_{g} T_{g,φ} 2 ^{g}φ ^{g}φ 0, \\ ^{g} ^{g}φ ^{g}φ _{g}^{2}φ. \end{matrix}  (13)

Integrating the trace of the first equation we obtain

0 _{M} Tr_{g} ^{g}φ _{g}^{2}g div_{g} T_{g,φ} 2 ^{g}φ ^{g}φ dv^{g} n 2 ^{g}φ _{g}^{2},

since the integral of Tr_{g} div_{g} T_{g,φ} vanishes by the divergence theorem. Hence ^{g}φ 0 for a critical point g, φ if n 3.

In addition, we can impose the (global) constraint vol M, g _{M} dv^{g} 1. For a constrained critical point g, φ the metric part of the gradient must be orthogonal to T_{g} _{1}, where _{1} is the space of unit volume metrics. In particular, T_{g} _{1} h Γ ^{2}T^{}M _{M} h, g _{g} dv^{g} 0 . Hence the right hand side of the first equation in (13) equals a constant multiple of the metric, i.e.

gφ _{g}^{2}g div T_{g,φ} 2 ^{g}φ ^{g}φ cg

for some c ℝ. Reasoning as above we obtain c \frac{n 2}{n} ^{g}φ _{g}^{2}, i.e. c 0 if n 2 and c 0 if n 3. In particular, if n 2 a constrained critical point is already a critical point. Summarising, we obtain

Corollary 4.10. Let n dim M 3. Then the following holds:

(i) g, φ is critical for if and only if ^{g}φ 0, i.e. φ is parallel with respect to g. In particular, the metric g is Ricci flat. Furthermore, any critical point is an absolute minimiser of . (ii) g, φ is critical for subject to the constraint vol M, g 1 if and only if

\begin{matrix}^{g}φ _{g}^{2}g div_{g}T_{g,φ} 2 ^{g}φ ^{g}φ cg, \\ ^{g} ^{g}φ ^{g}φ _{g}^{2}φ \end{matrix}

for some constant c 0. (A pair g, φ satisfying this system of equations will henceforth be called a spinor soliton.) If n 2, then g, φ is a genuine critical point for with vol M, g 1 if and only if it is a constrained critical point.

Example. (i) A parallel spinor not only implies that the underlying metric is Ricciflat, but also that the holonomy is a proper subgroup of SO n . The converse is also true for suitable choices of a spin structure [34], [39]. For example, in dimension 4 and 6 a parallel spinor forces the underlying Riemannian manifold to be Calabi-Yau (i.e. the holonomy is contained in SU 2 or SU 3 ), while in dimension 7 and 8 the holonomy is contained in G_{2} or Spin 7 . By Yau’s solution of the Calabi conjecture [44] and the work of Joyce on holonomy G_{2}- and Spin 7 -manifolds (see for instance his book [32]), compact examples, though not in an explicit manner, exist in abundance. (ii) A Killing spinor is defined to be a spinor satisfying the equation

_{X}^{g}φ λX φ

for all X Γ T M and some fixed λ ℝ. As a consequence of this equation g must be necessarily Einstein (see for instance [7]). A Killing spinor for a metric with vol M, g 1 is a constrained critical point for . Indeed, by the Killing equation

g ^{g}φ nλ^{2}φ ^{g}φ _{g}^{2}φ,

so that the second equation is satisfied. The symmetrisation of e_{i} e_{j} φ, _{e_{k}}^{g}φ is zero, for X φ, φ 0. Furthermore, ^{g}φ ^{g}φ λ^{2}g (again using the Killing equation), whence

gφ _{g}^{2}g div_{g} T_{g,φ} 2 ^{g}φ ^{g}φ n 2 λ^{2}g.

In particular, if n 2 a Killing spinor for a metric with vol M, g 1 is a genuine critical point for . (iii) More generally a twistor spinor is a spinor in the kernel of the Penrose- or twistor operator P_{g} Γ Σ_{g}M Γ ker μ defined by P_{g}φ p_{ker μ} ^{g}φ, where p_{ker μ} T^{}M Σ_{g}M ker μ denotes projection on the kernel of Clifford multiplication. In particular, any Killing spinor is a twistor spinor.

Proposition 4.11. If φ is a twistor spinor of unit norm on M^{n}, g with n 3, then it is a spinor soliton. Furthermore, the scalar curvature scal^{g} is constant.

Proof. By [7, Theorem I.3], P_{g}φ 0 implies

D_{g}^{2}φ \frac{n}{4 n 1} scal^{g}φ

such that using the Weitzenböck formula for D_{g}^{2}

g ^{g}φ D_{g}^{2}φ \frac{scal^{g}}{4} φ \frac{scal^{g}}{4 n 1} φ.

Hence, if g, φ is a twistor spinor of unit norm, then

gφ ^{2} ^{g} ^{g}φ, φ div_{g} ^{g}φ, φ \frac{scal^{g}}{4 n 1}

using ^{g}φ, φ \frac{1}{2}d φ, φ 0. It follows that ^{g} ^{g}φ ^{g}φ ^{2}φ, i.e. Q_{2} g, φ 0. To compute Q_{1} g, φ we recall that ψ X _{X}^{g}φ is unambigously defined for a unit vector field X [7, Theorem I.2 (4)]. Hence for an orthonormal basis e_{1}, …, e_{n}, we find _{e_{i}}^{g}φ, _{e_{j}}^{g}φ e_{i} ψ, e_{j} ψ which vanishes unless i j when it equals _{e_{i}}^{g}φ ^{2} ψ ^{2}. The contribution of ^{g}φ ^{g}φ is therefore just ^{g}φ ^{2}g n. It remains to compute the divergence term. If T̃ denotes the unsymmetrised 3, 0 - tensor we have

T̃_{ijk} e_{i} e_{j} φ, _{e_{k}}φ e_{i} e_{j} e_{k} φ, ψ .

Since X φ, ψ φ, _{X}φ for any unit vector X, the 3, 0 -tensor is totally skew, hence T_{g,φ} 0. Consequently,

Q_{1} g, φ \frac{2 n}{4n} ^{g}φ ^{2}g.

Now by the Bianchi identity 4.7, the divergence δ_{g}Q_{1} g, φ vanishes so that ^{g}φ ^{2} is constant. In particular, g, φ will be a spinor solition with c \frac{2 n}{4n} ^{g}φ ^{2} and constant scalar curvature scal^{g} 4 n 1 ^{g}φ ^{2}. □

4.3. The second variation. Next we investigate conditions for to be Morse-Bott, i.e. the critical set of forms a smooth manifold whose tangent bundle is precisely the kernel of the Hessian of (seen as an endomorphism via the L^{2}- metric). Towards this end, we calculate the second variation of at a critical point.

First, we linearise the spinor connection ^{g}φ as a function of g and φ. More formally, for X Γ T M consider the map

K_{X} , g, φ g, _{X}^{g}φ

and decompose the tangent space T_{g,φ} T_{g} _{g} as above.

Lemma 4.12. Let g, φ . Then D_{g,φ}K_{X} T_{g,φ} T_{g, _{X}^{g}φ} is given by

D_{g,φ}K_{X} ġ, φ̇ ġ, \frac{1}{4} _{i j} _{e_{i}}^{g} ġ X, e_{j} e_{i} e_{j} φ _{X}^{g} φ̇ .  (14)

Remark 4.13. Let α_{t} be a smooth curve of orientation preserving automorphisms of T M which are symmetric with respect to g_{0} g and such that α_{0} Id_{T M}. If g_{t} is the induced curve of metrics defined by g_{t} X, Y g α_{t}^{1} X, α_{t}^{1} Y , then ġ X, Y 2g α̇X, Y . Substituting this into (14) then gives Wang’s formula in [40, Proposition 1.5].

Proof. Again the vertical variation is the easy part for which we obtain

D_{g,φ}K_{X} 0, φ̇ 0, _{X}^{g} φ̇

with φ̇ Γ Σ_{g}M . Let now g_{t} be a path of metrics with g_{0} g and \frac{d}{dt} _{t 0} g_{t} ġ. Let g_{t}^{hor} be the horizontal lift of g_{t}. We may assume without loss of generality that g_{t} is a linear path, i.e. g_{t} g_{0} tġ. Then g_{t}^{hor} g_{t}, φ_{t} with φ_{t} B̂_{g_{t}}^{g}φ. Recall that if φ is locally expressed as b̃, φ̃ , then B̂_{g_{t}}^{g}φ is locally expressed as B̃_{g_{t}}^{g}b̃, φ̃ . With e_{i,t} B_{g_{t}}^{g} e_{i} the local spin frame B̃_{g_{t}}^{g}b̃ covers the local orthonormal frame B_{g_{t}}^{g}b e_{1,t}, …, e_{n,t} and we compute locally

D_{g,φ}K_{X} ġ, 0 \frac{d}{dt} _{t 0} _{X}^{g_{t}}φ_{t}
\frac{d}{dt} _{t 0} B̃_{g_{t}}^{g}b̃, Xφ̃ \frac{1}{2} _{i j} g_{t} _{X}^{g_{t}}e_{i,t}, e_{j,t} E_{i} E_{j} φ̃
\frac{d}{dt} _{t 0} B̃_{g_{t}}^{g}b̃, Xφ̃ \frac{1}{2} _{i j} g _{X}^{g}e_{i}, e_{j} E_{i} E_{j} φ̃

b̃, \frac{d}{dt} _{t 0} \frac{1}{2} _{i j} g_{t} _{X}^{g_{t}}e_{i,t}, e_{j,t} E_{i} E_{j} φ̃

\frac{d}{dt} _{t 0} B̂_{g_{t}}^{g} _{X}^{g}φ \frac{1}{2} _{i j} \frac{d}{dt} _{t 0} g_{t} _{X}^{g_{t}}e_{i,t}, e_{j,t} e_{i} e_{j} φ

using equation (2). The first term in this sum gives

\frac{d}{dt} _{t 0} B̂_{g_{t}}^{g} _{X}^{g}φ ġ, 0 T_{g, _{X}^{g}φ} .

For the second term we observe that

\frac{d}{dt} _{t 0} g_{t} _{X}^{g_{t}}e_{i,t}, e_{j,t} g \frac{d}{dt} _{t 0} _{X}^{g_{t}}e_{i}, e_{j} g _{X}^{g} \frac{d}{dt} _{t 0} e_{i,t}, e_{j}

if we compute at a point x M with an orthonormal frame such that ^{g}e_{i} x 0. Using the generalised cylinder calculus as in the proof of Theorem 4.8 we get

\frac{d}{dt} _{t 0} e_{i,t} ν, e_{i,t} _{t 0}  _{e_{i,t}}^{C}ν _{t 0} W e_{i}  \frac{1}{2} ġ e_{i}, ,

since _{ν}^{C}e_{i,t} 0 and W X ġ X, 2 by (14) in [6]. Consequently,

g _{X}^{g} \frac{d}{dt} _{t 0} e_{i,t}, e_{j} \frac{1}{2} _{X}^{g} ġ e_{i}, e_{j} .

Furthermore, by Theorem 1.174 (a) in [8] we have

g \frac{d}{dt} _{t 0} _{X}^{g_{t}}e_{i}, e_{j}  \frac{1}{2}  _{X}^{g} ġ e_{i}, e_{j}  _{e_{i}}^{g} ġ X, e_{j}  _{e_{j}}^{g} ġ e_{i}, X ,

whence

\frac{1}{2} _{i j} \frac{d}{dt} _{t 0} g_{t} _{X}^{g_{t}}e_{i,t}, e_{j,t} e_{i} e_{j} φ \frac{1}{4} _{i j} _{e_{i}}^{g} ġ X, e_{j} e_{i} e_{j} φ.

This is the asserted formula. □

For g, φ let κ_{g,φ} T_{g,φ} Γ T^{}M Σ_{g}M be the map defined by

κ_{g,φ} ġ, φ̇ \frac{1}{4} _{i j} _{e_{i}}^{g} ġ , e_{j} e_{i} e_{j} φ ^{g}φ̇,

so that D_{g,φ}K_{X} ġ, φ̇ ġ, κ_{g,φ} ġ, φ̇ X . Take a smooth path g_{t}, φ_{t} with g_{0}, φ_{0} g, φ and write ^{g_{t}}φ_{t} _{i} e_{i}^{}_{e_{i}}^{g_{t}}φ_{t} for a local orthonormal frame e_{i} with respect to g. Then we have

\frac{d}{dt} _{t 0} ^{g_{t}}φ_{t} _{g_{t}}^{2} \frac{d}{dt} _{t 0} _{i,j} g_{t} e_{i}^{}, e_{j}^{}_{e_{i}}^{g_{t}}φ_{t}, _{e_{j}}^{g_{t}}φ_{t}

_{i,j} ġ e_{i}, e_{j} _{e_{i}}^{g}φ, _{e_{j}}^{g}φ 2 _{i} κ_{g,φ} ġ, φ̇ e_{i} , _{e_{i}}^{g}φ

ġ, ^{g}φ ^{g}φ _{g} 2 κ_{g,φ} ġ, φ̇ , ^{g}φ _{g}  (15)

with ġ \frac{d}{dt} _{t 0} g_{t} and φ̇ \frac{d}{dt} _{t 0} φ_{t} (recall that \frac{d}{dt} _{t 0} g_{t} v^{}, w^{}ġ v, w for any v, w T_{x}M).

Proposition 4.14 (First variation revisited). Let g_{t}, φ_{t} be a smooth path in with g_{0}, φ_{0} g, φ . Then

\frac{d}{dt} _{t 0} g_{t}, φ_{t}  _{M} ġ, \frac{1}{4} ^{g}φ _{g}^{2}g \frac{1}{4} div_{g} T_{g,φ} \frac{1}{2} ^{g}φ ^{g}φ _{g} dv^{g}
M φ̇, ^{g} ^{g}φ _{g} dv^{g}

with ġ \frac{d}{dt} _{t 0} g_{t} and φ̇ \frac{d}{dt} _{t 0} φ_{t}.

Proof. Using equation (15) we compute

\frac{d}{dt} _{t 0} g_{t}, φ_{t}

\frac{1}{2} _{M} ^{g}φ _{g}^{2} \frac{d}{dt} _{t 0} dv^{g_{t}}  \frac{1}{2} _{M} \frac{d}{dt} _{t 0} ^{g_{t}}φ_{t} _{g_{t}}^{2} dv^{g}
\frac{1}{4} _{M} ^{g}φ _{g}^{2} Tr_{g}ġ dv^{g} \frac{1}{2} _{M} ġ, ^{g}φ ^{g}φ _{g} dv^{g}  _{M} κ_{g,φ} ġ, φ̇ , ^{g}φ _{g} dv^{g}.

Now

\frac{1}{4} _{M} ^{g}φ _{g}^{2} Tr_{g}ġ dv^{g} \frac{1}{4} _{M} ^{g}φ _{g}^{2} ġ, g _{g} dv^{g}

and

κ_{g,φ} ġ, φ̇ , ^{g}φ _{g}  \frac{1}{4} _{i j} _{k}  _{e_{i}}^{g} ġ e_{k}, e_{j} e_{i} e_{j} φ, _{e_{k}}^{g}φ _{k} _{e_{k}}^{g} φ̇, _{e_{k}}^{g}φ
\frac{1}{4} ^{g}ġ, T_{g,φ} _{g} ^{g}φ̇, ^{g}φ _{g}.

Since

\frac{1}{4} _{M} ^{g}ġ, T_{g,φ} _{g} dv^{g} \frac{1}{4} _{M} ġ, div_{g} T_{g,φ} _{g} dv^{g}

and

M ^{g}φ̇, ^{g}φ _{g} dv^{g} _{M} φ̇, ^{g} ^{g}φ _{g} dv^{g}

the result follows. □

Proposition 4.15 (Second variation). Let g, φ with ^{g}φ 0 and let g_{t}, φ_{t} be a smooth path in with g_{0}, φ_{0} g, φ . Then

\frac{d^{2}}{dt^{2}} _{t 0} g_{t}, φ_{t}  _{M} κ_{g,φ} ġ, φ̇ _{g}^{2} dv^{g}  _{M} κ_{g,φ}^{}κ_{g,φ} ġ, φ̇ , ġ, φ̇ _{g} dv^{g} 0

with equality if and only if κ_{g,φ} ġ, φ̇ 0.

Proof. Computing as before we get

\frac{d^{2}}{dt^{2}} _{t 0} g_{t}, φ_{t}  \frac{d}{dt} _{t 0} \frac{1}{4} _{M} ^{g_{t}}φ_{t} _{g_{t}}^{2} Tr_{g_{t}} ġ_{t} dv^{g_{t}}  \frac{1}{2} _{M} ġ_{t} ^{g_{t}}φ_{t}, ^{g_{t}}φ_{t} dv^{g_{t}}
M κ_{g_{t},φ_{t}} ġ_{t}, φ̇_{t} , ^{g_{t}}φ_{t} _{g_{t}} dv^{g_{t}}
M κ_{g,φ} ġ, φ̇ _{g}^{2} dv^{g}

since any ^{g_{t}}φ_{t}-term which is not differentiated vanishes by assumption when evaluated at t 0. For instance, _{i} \frac{d}{dt} _{t 0} e_{i,t} _{e_{i,t}}^{g_{t}}φ_{t} _{i} e_{i} \frac{d}{dt} _{t 0} _{e_{i,t}}^{g_{t}}φ_{t} κ_{g,φ} ġ, φ̇ . The result follows. □

The linearisation L_{g,φ} D_{g,φ}Q T_{g,φ} T_{g,φ} of the negative gradient of at a critical point is just the negative Hessian of regarded as an endomorphism via the L^{2}-metric.

Corollary 4.16. If g, φ satisfies ^{g}φ 0, then

L_{g,φ} π^{T} κ_{g,φ}^{}κ_{g,φ}.

In particular, L_{g,φ} is formally selfadjoint and non-positive in the sense that

L_{g,φ} ġ, φ̇ , ġ, φ̇ _{g} 0

for all ġ Γ ^{2}T^{}M and φ̇ Γ Σ_{g}M . Furthermore, ker L_{g,φ} ker κ_{g,φ}, which is the space of infinitesimal deformations preserving a parallel spinor.

Let Crit denote the critical set of and let be the space of Ricci-flat metrics. In the following we assume n dim M 3, which implies that g, φ is in Crit if and only if ^{g}φ 0. In particular, g for g, φ Crit .

Recall that a Riemannian manifold M, g is irreducible if its universal Riemannian cover is not isometric to a Riemannian product. A critical point will be called irreducible if the underlying Riemannian manifold is irreducible.

Theorem 4.17. Let n dim 4 and ḡ, φ̄ be an irreducible critical point. If either (i) M is simply-connected, or (ii) n is not divisible by four, or (iii) n 4 or 12, then there exists a smooth neighbourhood of ḡ, φ̄ inside Crit such that T_{ḡ,φ̄} ker L_{ḡ,φ̄}.

Remark 4.18. In dimension 3 a metric with a parallel spinor is necessarily flat and in fact only flat tori can carry such metrics [36].

Before we actually prove this theorem we first remark that by Corollary 4.16 and Remark 4.13, ker L_{g,φ} ker d ^{0} for g, φ Crit , where d ^{0} is defined as in [40, Proposition 2.2]. In particular, we get

ker L_{g,φ} δ_{g}^{1} 0 Tr^{1} 0  ġ, φ̇ δ_{g}ġ 0, Tr_{g}ġ 0, ^{g}φ̇ 0, _{g}Ψ_{ġ,φ} 0 . (16)

The twisted Dirac operator _{g} Γ T^{}M Σ_{g}M Γ T^{}M Σ_{g}M is defined locally by _{g} α φ e_{k} α _{e_{k}}^{g}φ α D_{g}φ, and is evaluated on Ψ_{ġ,φ}, the spinor-valued 1-form given by Ψ_{ġ,φ} X ġ X φ. By [40, Proposition 2.10], [13, Proposition 2.4], gΨ_{ġ,φ} 0 if and only if Δ_{L}ġ 0. Here, Δ_{L} denotes the Lichnerowicz Laplacian defined with respect to g. Its kernel consists of the transverse traceless infinitesimal deformations of g [8, Theorem 1.174 (d)]. It follows that the space on the right hand side of (16) can be identified with ker Δ_{L} g-parallel spinors . Smoothness at g, φ Crit will therefore follow from two properties. For the Diff_{0} M -action on we need a smooth slice through g which gives rise to a smooth open neighbourhood of g. Then ker L_{g,φ} can be identified with T_{g} g-parallel spinors . Secondly, D_{g}^{2} ^{g} ^{g} scal^{g} 4 in virtue of the Weitzenböck formula, so that any g-harmonic spinor must be parallel for g . We therefore consider the restriction of the fibre bundle to together with the “universal” Dirac operator D _{}_{}which sends Φ g, φ to g, D_{g}φ . Formally, this is a smooth family of elliptic operators. If dim ker D^{g} is constant on , then _{g} ker D_{g} forms a smooth vector bundle over and we may take to be the total space of the associated unit sphere bundle.

Proof of Theorem 4.17. Under the assumption of the theorem it follows from [39] for case (i) and [41] (building on McInnes work [33]) for case (ii) and (iii) that the holonomy must be equal to either G_{2}, Spin 7 , SU m or Sp k depending on n and the number of linearly independent spinors. Consequently, g defines a torsionfree G-metric in the sense of [35]. As shown there, these metrics define an open smooth subset of as a consequence of Goto’s unobstructedness theorem [22].

To settle the local constancy we shall proceed case by case, using the following facts. First assume n dim M to be even so that we have a decomposition _{g} _{g} _{g} into positive and negative spinors. The index of D_{g}^{}D_{g} _{_{g}} is given by ind D_{g}^{} δ_{g}^{}δ_{g}^{}, where δ_{g}^{}dim ker D_{g}^{}. The δ_{g}^{}are upper semi-continuous in g so they can only decrease simultaneously in a neighbourhood of ḡ, for the index is locally constant. In particular, δ_{g}^{}is locally constant if either δ ; ḡ or δ ; ḡ vanishes. On the other hand ḡ is irreducible by assumption, and so are metrics close to ḡ as this is an open condition. Consequently, δ ; ḡ can be read off from the classification results of Wang [39] if M is simply-connected and [41] for the non-simply connected case. If M is simply-connected there are four cases if n is even: Spin 7 (n 8), SU m for n 2m even or odd respectively, and finally Sp m for n 4m. The only non-trivial case where both δ ; ḡ and δ ; ḡ are non-zero occurs for SU m , m odd. Here δ ; ḡ δ ; ḡ 1. But if δ_{g}^{}were not constant near ḡ, then necessarily δ_{g}^{}δ_{g}^{}0 arbitrarily close to ḡ. This cannot happen for irreducible Ricci-flat metrics of special holonomy are open in . Finally, if n is odd, the only case which can occur is holonomy G_{2}. Again any Ricci-flat metric close enough to ḡ must have holonomy exactly G_{2} so that dim ker D_{g} 1. The non simply-connected case follows analogously if one excludes the case of holonomy SU 4m ℤ_{2} for n 8m and Sp k ℤ_{d}, where d 1 odd divides m 1. □

Remark 4.19. In any case it follows from Wang’s stability theorem [40, Theorem 3.1] and [41, Section 3] in conjunction with [22] and [35] that the irreducible metrics with parallel spinor form an open (but not necessarily smooth) subset of on which ker D is locally constant.

In low dimensions, irreducibility can be enforced by the topology of the underlying manifold. Indeed, we have the

Lemma 4.20. Let M be a compact spin manifold. Assume either that

(i) n dim M 4, 6 or 7 and M has finite fundamental group, or (ii) n 8, M is simply-connected and not a product of two K3-surfaces. Then any metric which admits a parallel spinor is irreducible.

Corollary 4.21. Under the above assumptions, the functional is Morse–Bott, i.e. the critical set of is a submanifold with tangent space precisely given by the kernel of the Hessian of .

Proof of Lemma 4.20. If M^{n}, g admits a parallel spinor, then M^{n} has a finite Riemannian cover of the form M̃^{n k} , g̃ T^{k}, g_{0} where M̃^{n k} , g̃ is a compact simply-connected Ricci-flat manifold [11], [17]. Since π_{1} M is finite this implies that k 0, identifying M̃ with the universal covering of M. In dimensions 4, 6 and 7 we conclude that M̃ is irreducible for otherwise its de Rham decomposition would contain a Euclidean factor on dimensional grounds. In dimension 8, if M is simply-connected and admits a parallel spinor, the holonomy group of M is either SU 2 SU 2 , Sp 2 , SU 4 or Spin 7 , where M is irreducible in the latter three cases and is a product of two K3-surfaces in the first case. □

Remark 4.22. In dimension 5 a Riemannian manifold with a parallel spinor is either flat or a mapping torus of a holomorphic isometry of a K3-surface [20].

Remark 4.23. In an article in preparation [3] the authors prove in collaboration with Klaus Kröncke that dim ker D is locally constant on the space of metrics admitting a nontrivial parallel spinor. In particular, if n 3 and ḡ, φ̄ is a critical point, then there exists a smooth neighbourhood of ḡ, φ̄ inside Crit such that T_{ḡ,φ̄} ker L_{ḡ,φ̄}.

5. The negative gradient flow

In order to find critical points of the energy functional , it is natural to consider its negative gradient flow on .

Definition 5.1. Let Φ g, φ . We call a solution to

\frac{∂}{∂t}Φ_{t} grad Φ_{t} , Φ_{0} Φ  (17)

the spinor flow with initial condition Φ or the spinor flow for short.

The main goal of this section is to show that the spinor flow equation (17) has a unique short-time solution, i.e. to prove the following

Theorem 5.2. Given Φ g, φ , there exists ε 0 and a smooth family Φ_{t} for t 0, ε such that

\frac{∂}{∂t}Φ_{t} grad Φ_{t} , Φ_{0} Φ.

Furthermore, if Φ_{t} and Φ_{t}^{}are solutions to (17), then Φ_{t} Φ_{t}^{}whenever defined. Hence Φ_{t} is uniquely defined on a maximal time-interval 0, T for some 0 T .

5.1. Existence. In this subsection we establish the existence part. As a first step, we note the following

Lemma 5.3. The operator Q T is a second order, quasilinear differential operator.

Proof. The assertion is a straightforward consequence of Theorem 4.8. Only the rough Laplacian ^{g} ^{g}φ and the divergence div_{g} T_{g,φ} contribute to the highest order terms. Expressed in a local basis, these are of the form a φ, φ _{e_{j}} _{e_{k}} for a smooth function a depending on the 1-jet of φ. □

In order to apply the standard theory of quasilinear parabolic equations (see for instance [4, §4.4.2], [30, §7.8], [37, §7.1] or [38, §4]) we recall the following definition. Consider a Riemannian vector bundle E, , . Let Q_{t} Γ E Γ E be a timedependent family of second order quasilinear differential operators. We say that this family is strongly elliptic at u_{0} if there exists a constant λ 0 such that the linearisation D_{u_{0}}Q_{0} of Q_{0} at u_{0} satisfies

σ_{ξ} D_{u_{0}}Q_{0} v, v λ ξ ^{2} v ^{2}  (18)

for all ξ T_{x}^{}M, ξ 0, and v E_{x}. If equation (18) only holds with λ 0, we say that Q_{t} is weakly elliptic at u_{0}. Note that we define the principal symbol of a linear operator L as

σ_{ξ} L v \frac{i^{k}}{k!} L f^{k}u x

for any f C^{}M with f x 0, df_{x} ξ and u Γ E with u x v which accounts for the minus sign in (18). The corresponding flow equation

\frac{∂}{∂t}u Q_{t} u , u 0 u_{0}  (19)

will be called strongly parabolic at u_{0} or weakly parabolic at u_{0} respectively. In case of strong parabolicity we have short-time existence and uniqueness for the flow equation (19).

Observe that in our case Q grad is a map from Γ S ΣM to T , so that formally speaking we are not in the situation just described. However, using the natural connection of Section 3, a neighbourhood of a section Φ g, φ Γ S ΣM may be identified with a neighbourhood of the 0-section in the vector bundle ^{2}T^{}M φ^{}. More precisely, we project φ̇_{x} φ^{}onto S Σ_{g}M using the fiberwise exponential map and parallel translate the resulting unit spinor along the linear path g_{x} tġ_{x} for ġ_{x} ^{2}T^{}M. We may thus identify sections of S ΣM close to Φ with sections ġ, φ̇ of Γ ^{2}T^{}M Γ φ^{}accordingly. By a slight abuse of notation, we can therefore consider Q as well as its linearisation D_{Φ}Q as an operator from Γ ^{2}T^{}M Γ φ^{}to itself.

First we compute the principal symbol of Q, i.e. the principal symbol of its linearisation D_{Φ}Q. Let X Γ T M be a vector field. Recall from Lemma 4.12 that the linearisation of the spinor connection K_{X} , g, φ g, _{X}^{g}φ is given by

D_{g,φ}K_{X} ġ, φ̇ ġ, \frac{1}{4} _{i j} _{e_{i}}^{g} ġ X, e_{j} e_{i} e_{j} φ _{X}^{g} φ̇ .

Lemma 5.4. Let 0 ξ T_{x}^{}M. The principal symbol of the linearisation of the spinor connection at ξ is given by

σ_{ξ} D_{g,φ}K_{X} ġ_{x}, φ̇_{x} 0, \frac{i}{4}ξ ġ_{x} X, φ iξ X φ̇_{x} .

Proof. Let f be a smooth function on M with f x 0 and df_{x} ξ and ġ be a symmetric 2-tensor extending ġ_{x}, resp. φ̇ a spinor extending φ̇_{x}. Then

σ_{ξ} D_{g,φ}K_{X} ġ_{x}, φ̇_{x} i D_{g,φ}K_{X} fġ, fφ̇ x

0, \frac{i}{4} _{i j} df e_{i} ġ X, e_{j} e_{i} e_{j} φ idf X φ̇ x ,

whence the result. □

We can now proceed and compute the principal symbol of D_{g,φ}Q.

Proposition 5.5. Let 0 ξ T_{x}^{}M and e_{1} ξ^{}ξ , e_{2}, …, e_{n} be an orthonormal basis of T_{x}M. Further, define β_{ξ} _{j 1}^{n} ξ e_{j} φ, φ̇_{x} e_{j}. Then the principal symbol of the linearisation of Q at ξ is given by

σ_{ξ} D_{g,φ}Q ġ_{x}, φ̇_{x}  { \atop }\begin{matrix}\frac{1}{16} ξ ^{2}ġ_{x} ξ ġ_{x} ξ,  \frac{1}{4}ξ β_{ξ} \\ \frac{1}{4}ξ ġ_{x} ξ, φ ξ ^{2}φ̇_{x} \end{matrix} { \atop }.

Proof. We first notice that in order to compute the principal symbol we only need to linearise the highest order terms in Q, i.e. the term \frac{1}{4} div_{g} T_{g,φ} in Q_{1} and the term ^{g} ^{g}φ in Q_{2}. In the following we write for equality up to terms of lower order (one or zero). Firstly, beginning with Q_{2}, we get

D_{g,φ}Q_{2} ġ, φ̇ D_{g,φ} g, φ  ^{g} ^{g}φ ġ, φ̇

_{i} _{e_{i}}^{g} D_{g,φ}K_{e_{i}} ġ, φ̇ ,

and the usual symbol calculus together with Lemma 5.4 yields

σ_{ξ} D_{g,φ}Q_{2} ġ_{x}, φ̇_{x}  _{i 1}^{n} ξ e_{i} \frac{1}{4}ξ ġ_{x} e_{i}, φ ξ e_{i} φ̇_{x}
\frac{1}{4}ξ ġ_{x} ξ, φ ξ ^{2}φ̇_{x}.

In a similar fashion, we obtain

D_{g,φ}Q_{1} ġ, φ̇ \frac{1}{4}D_{g,φ} g, φ div_{g} T_{g,φ} ġ, φ̇

\frac{1}{4} _{i 1}^{n} _{j,k 1}^{n} e_{i} e_{i} e_{j} φ, D_{g,φ}K_{e_{k}} ġ, φ̇ e_{j} e_{k}

\frac{1}{4} _{i,j,k} e_{i} e_{j} φ, _{e_{i}}^{g} D_{g,φ}K_{e_{k}} ġ, φ̇ e_{j} e_{k},

whence

σ_{ξ} D_{g,φ}Q_{1} ġ_{x}, φ̇_{x}
\frac{1}{4} _{i,j,k} e_{i} e_{j} φ, ξ e_{i} \frac{1}{4}ξ ġ_{x} e_{k}, φ ξ e_{k} φ̇_{x} e_{j} e_{k}
_{j,k} \frac{1}{16} ξ e_{j} φ, ξ ġ_{x} e_{k}, φ \frac{1}{4} ξ e_{j} φ, ξ e_{k} φ̇_{x} e_{j} e_{k}.

The second term in the last line contributes ξ β 4. Using again the general rule v ξ^{}v ξ^{}ξ v the first term gives

\frac{1}{16} _{j,k} ξ e_{j} φ ξ e_{j} φ, ξ ġ_{x} e_{k}, φ ġ_{x} e_{k}, ξ φ e_{j} e_{k}

\frac{1}{16} _{j,k} ξ e_{j} φ, ξ ġ_{x} e_{k}, φ ξ e_{j} φ, ġ_{x} e_{k}, ξ φ
ξ e_{j} φ, ξ ġ_{x} e_{k}, φ ξ e_{j} φ, ġ_{x} e_{k}, ξ φ e_{j} e_{k}
\frac{φ ^{2}}{16} ξ ^{2}ġ_{x} ξ ġ_{x} ξ, ξ ġ_{x} ξ, ξ ġ_{x} ξ,
\frac{φ ^{2}}{16} ξ ^{2}ġ_{x} ξ ġ_{x} ξ, .

When passing from the second to the third line, we use the skew-adjointness of the Clifford multiplication and in particular the relation v φ, w φ g v, w φ ^{2}. Since φ 1 the desired formula follows. □

Corollary 5.6. The principal symbol of D_{g,φ}Q is negative semi-definite, i.e.

g σ_{ξ} D_{g,φ}Q ġ_{x}, φ̇_{x} , ġ_{x}, φ̇_{x} 0

for all ξ T_{x}^{}M, ġ_{x} ^{2}T_{x}^{}M and φ̇_{x} φ_{x}^{}. For ξ T_{x}^{}M with ξ 1 the kernel of the symbol is given by

ker σ_{ξ} D_{g,φ}Q  ġ_{x}, φ̇_{x} ġ_{x} v, w 0 for v, w ξ^{}, φ̇_{x}  \frac{1}{4}ξ ġ_{x} ξ, φ ,

which is an n-dimensional subspace of ^{2}T_{x}^{}M φ_{x}^{}.

Proof. By homogeneity of the symbol we may assume that ξ 1 to begin with. Furthermore, we decompose ġ_{x} aξ ξ ξ α α ξ γ, i.e. (using the basis of the previous proposition) α _{k 2}^{n} ġ_{x} ξ, e_{k} e_{k} and γ _{j,k 2}^{n} ġ_{x} e_{j}, e_{k} e_{j} e_{k}. Then

g σ_{ξ} D_{g,φ}Q ġ_{x}, φ̇_{x} , ġ_{x}, φ̇_{x}
\frac{1}{16} ġ_{x} ^{2} \frac{1}{16} ξ ġ_{x} ξ, , ġ_{x}  \frac{1}{4} ξ β_{ξ}, ġ_{x}  \frac{1}{4} ξ ġ_{x} ξ, φ, φ̇ φ̇_{x} ^{2}
\frac{1}{16} a^{2} 2 α ^{2} γ ^{2}  \frac{1}{16} α ^{2} a^{2}  \frac{1}{4}g α, β_{ξ}  \frac{1}{4}g α, β_{ξ}  φ̇_{x} ^{2}
\frac{1}{16} α ^{2} \frac{1}{16} γ ^{2} \frac{1}{2}g α, β_{ξ}  φ̇_{x} ^{2},
since g α, β_{ξ}  _{j} α_{j} ξ e_{j} φ, φ̇_{x} ξ α φ, φ̇_{x} . This also gives g α, β_{ξ}  α φ̇_{x}

with equality if and only if φ̇_{x} is a multiple of ξ α φ. Young’s inequality implies that α ^{2} 8 α φ̇_{x} 16 φ̇_{x} ^{2} 0 with equality if and only if α 4 φ̇_{x} . From there the computation of the kernel is straightforward. □

The kernel of σ_{ξ} D_{g,φ}Q reflects the diffeomorphism invariance of the equation. More specifically, consider the principal symbol of the linear operator

λ_{g,φ}^{}Γ T M Γ ^{2}T^{}M Γ Σ_{g}M
X _{X}g, ˜ _{X}^{g} φ 2δ_{g}^{}X^{}, _{X}^{g}φ \frac{1}{4}dX^{}φ .

Using σ_{ξ} X 2δ_{g}^{}X^{}v i ξ v^{}v^{}ξ and σ_{ξ} X _{X}^{g}φ \frac{1}{4}dX^{}φ v \frac{i}{4}ξ v^{}φ we get

σ_{ξ} λ_{g,φ}^{}v i ξ v^{}v^{}ξ, \frac{1}{4}ξ v^{}φ

and thus im σ_{ξ} λ_{g,φ}^{}ker σ_{ξ} D_{g,φ}Q . In fact, if g, φ is a critical point of and L_{g,φ} D_{g,φ}Q, then the associated “deformation complex”

0 Γ T M ^{λ_{g,φ}^{}}T_{g,φ} ^{L_{g,φ}} T_{g,φ} ^{λ_{g,φ}} Γ T M 0

is elliptic. This may easily be checked using the Bianchi identity (Corollary 4.7) and the formal self-adjointness of L_{g,φ}.

We remedy this situation by adding an additional term to Q in order to break the diffeomorphism invariance. Towards that end, we define a new operator

Q Φ Q Φ λ_{Φ}^{}X Φ

for a vector field X Φ depending on the 1-jet of Φ g, φ . Then λ_{g,φ}^{}X Φ depends on the 2-jet of Φ and modifies the highest order terms in Q. This procedure is known as DeTurck’s trick and was initially applied to give an alternative proof of short-time existence and uniqueness of Ricci flow, cf. [14]. A suitable vector field can be determined as follows. Let ḡ be a background metric. We define the linear operator

Xḡ Γ T M , g 2 δḡg ^{}, (20)

where ^{}is taken with respect to the fixed metric ḡ.

Definition 5.7. Let ḡ be a fixed background metric and set

{ \atop Qḡ} T , g, φ Q g, φ λ_{g,φ}^{}Xḡ g .

We call a solution to

\frac{∂}{∂t} Φ̃_{t} { \atop Qḡ} Φ̃_{t} , Φ̃_{0} Φ̄

the gauged spinor flow associated with ḡ and with initial condition Φ̄ ḡ, φ̄ or gauged spinor flow for short.

Proposition 5.8. Let ḡ be a fixed background metric. Then the gauged spinor flow associated with ḡ is strongly parabolic at ḡ, φ̄ for any φ̄ ḡ.

Proof. We need to compute the principal symbol of D_{ḡ,φ̄} { \atop Qḡ} and check that the quadratic form

ġ_{x}, φ̇_{x} ḡ σ_{ξ} D_{ḡ,φ̄} { \atop Qḡ} ġ_{x}, φ̇_{x} , ġ_{x}, φ̇_{x}

is negative definite. The principal symbol of D_{ḡ,φ̄}Q is known by Proposition 5.5. The linearisation of the map g _{Xḡ g}g at ḡ is given by

Dḡ g _{Xḡ g}g ġ  _{Xḡ ġ} ḡ terms of lower order in ġ

4δ ; ḡδḡġ terms of lower order in ġ

where we have inserted the definition of Xḡ ġ as in (20). Now for a 1-form ζ T_{x}^{}M

one has σ_{ξ} δ ; ḡ ζ \frac{i}{2} ξ ζ ζ ξ and σ_{ξ} δḡ ġ_{x} iġ_{x} ξ, . Hence

σ_{ξ} Dḡ g _{Xḡ g}g ġ_{x} 2 ξ ζ ζ ξ 4ξ ζ,

where as above ζ ġ_{x} ξ, aξ α. On the other hand, the linearisation of the map g, φ _{ X̃ḡ g} φ at ḡ, φ̄ is given by

D_{ḡ,φ̄} g, φ _{  X̃ḡ g} φ ġ, φ̇ _{  X̃ḡ ġ} φ̄ terms of lower order in ġ, φ̇
\frac{1}{2}dδḡġ φ̄ terms of lower order in ġ, φ̇

and therefore

σ_{ξ} D_{ḡ,φ̄} g, φ _{  X̃ḡ g} φ ġ_{x}, φ̇_{x}  \frac{1}{2}ξ ġ_{x} ξ, φ̄.

It follows by the previous computations of Corollary 5.6 that

ḡ σ_{ξ} D_{ḡ,φ̄}Qḡ ġ_{x}, φ̇_{x} , ġ_{x}, φ̇_{x}
\frac{1}{16} α ^{2} \frac{1}{16} γ ^{2} \frac{1}{2} ḡ α, β_{ξ}  φ̇_{x} ^{2} 4a^{2} 4 α ^{2} \frac{1}{2} ξ ġ_{x} ξ, φ̄, φ̇_{x}
4a^{2} \frac{65}{16} α ^{2} \frac{1}{16} γ ^{2} φ̇_{x} ^{2}  \frac{1}{16} ġ_{x} ^{2} φ̇_{x} ^{2},

where we used the relation ḡ α, β_{ξ} ξ α φ̄, φ̇_{x} . This proves the assertion. □

Finally, we are in a position to establish short-time existence of the spinor flow for a given arbitrary initial condition Φ̄ ḡ, φ̄ . Consider the (unique) solution Φ̃_{t} g̃_{t}, φ̃_{t} to the gauged spinor flow equation associated with ḡ subject to the same initial condition Φ̄. Let f_{t} be the 1-parameter family f_{t} in Diff_{0} M determined by the (non-autonomous) ordinary differential equation

\frac{d}{dt}f_{t} Xḡ g̃_{t} f_{t}, f_{0} id_{M} .  (21)

Then f_{t} lifts to a 1-parameter family F_{t} in { \atop Diff}_{s} M with F_{0} idP̃ and we claim that Φ_{t} F_{t}^{}Φ̃_{t} is a solution to the spinor flow equation with initial condition Φ̄. Indeed, we have Φ_{0} F_{0}^{}Φ̃_{0} Φ̄. Furthermore, by Lemma 9 and the equivariance properties established in Proposition 4.3 we obtain

\frac{∂}{∂t}Φ_{t} \frac{∂}{∂t}F_{t}^{}Φ̃_{t}

F_{t}^{}{ \atop Qḡ} Φ̃_{t} _{Xḡ g̃_{t}} g̃_{t}, ˜ _{Xḡ g̃_{t}}^{g} φ̃_{t}

F_{t}^{}Q Φ̃_{t} Q Φ_{t} .

This proves existence.

5.2. Uniqueness. The proof of uniqueness is modelled on the proof in the Ricci flow case given by Hamilton in [24]. A similar argument has also been applied in [42] for the Dirichlet energy flow on the space of positive 3-forms on a 7-manifold. In the existence proof g_{t} was obtained from g̃_{t} and the solution f_{t} of the ODE (21) by setting g_{t} f_{t}^{}g̃_{t}. Conversely, substituting g̃_{t} f_{t}g_{t} turns (21) into the PDE

\frac{d}{dt}f_{t} Xḡ f_{t}g_{t} f_{t}, f_{0} id_{M} .  (22)

To show uniqueness of the spinor flow we want to use uniqueness of the gauged spinor flow. To that end we pass from a spinor flow solution to a gauged spinor flow solution by solving (22). For Riemannian metrics g and ḡ let

P_{g,ḡ} C^{}M, M T C^{}M, M , f df X_{f^{}ḡ} g .

Note that P_{g,ḡ} f Γ f^{}T M T_{f}C^{}M, M so that P_{g,ḡ} may be viewed as a section of the bundle T C^{}M, M C^{}M, M . Furthermore, for a diffeomorphism f we recover df X_{f^{}ḡ} g Xḡ f_{}g f as above.

Lemma 5.9. The operator P_{g,ḡ} is a second order, quasilinear differential operator.

Proof. The proof essentially amounts to a calculation in local coordinates, cf. the proof of Lemma 6.1 in [42]. Further details are left to the reader. □

The proof of existence of solutions to (22) resembles the short-time existence proof of the spinor flow. First, using the exponential map on M with respect to the metric ḡ we may identify maps close to a fixed map f with sections of the bundle f^{}T M. Furthermore, by parallel translating along radial geodesics, we may assume that P_{g,ḡ} takes values in the same fixed vector bundle f^{}M. Hence P_{g,ḡ} as well as its linearisation D_{f}P_{g,ḡ} can be considered as operators Γ f^{}T M Γ f^{}T M .

Lemma 5.10. The linearisation of P_{ḡ,ḡ} at f id_{M} is given by

D_{id_{M}} P_{ḡ,ḡ} Γ T M Γ T M , X 4 δḡδ ; ḡX^{}^{},

where ^{}and ^{}are taken with respect to ḡ.

Proof. Let f_{t} be the flow of X Γ T M . Fix x M and let τ ḡ ; t denote the parallel transport induced by the Levi-Civita connection ḡ along the path t f_{t} x with inverse τ ḡ ; t. Then

\frac{d}{dt} _{t 0} τ ḡ ; t δḡ f_{t} ḡ ^{}f_{t} x  \frac{d}{dt} _{t 0} τ ḡ ; t δḡ f_{t}^{}ḡ ^{}f_{t} x

ḡ ; X δḡ ḡ ^{}x δḡ _{X} ḡ ^{}x

2 δḡδ ; ḡX^{}^{}x ,

since δḡḡ 0 and _{X} ḡ 2δ ; ḡX^{}. Taking into account that Xḡ g 2 δḡg ^{}, the asserted formula follows. □

Proposition 5.11. Let g_{t} be a smooth family of Riemannian metrics with g_{0} ḡ. Then the mapping flow \frac{d}{dt}f_{t} P_{g_{t},ḡ} is strongly parabolic at t 0 and f_{0} id_{M} and hence has a unique short time solution with initial condition f_{0} id_{M}.

Proof. The symbol of 4δḡδ ; ḡ Γ T^{}M Γ T^{}M evaluated in ξ T_{x}^{}M with ξ 1 is given by

4σ_{ξ} δḡδ ; ḡ ζ 2 ζ ḡ ζ, ξ ξ

for any ζ T_{x}^{}M. Hence

4ḡ σ_{ξ} δḡδ ; ḡ ζ , ζ 2ḡ ζ, ζ 2ḡ ζ, ξ ^{2} 2 ζ 2 ; ḡ

for any ζ T_{x}^{}M. □

Remark 5.12. The solution stays a diffeomorphism for short time since Diff M is open in C^{}M, M .

The proof of uniqueness can now be finished as follows: Suppose that Φ_{t} g_{t}, φ_{t} and Φ_{t}^{}g_{t}^{}, φ_{t}^{}are two spinor flow solutions on 0, ε with the same initial condition Φ̄ ḡ, φ̄ . Solving (22) with g_{t} and g_{t}^{}gives two families of diffeomorphisms f_{t} and f_{t}^{}which we assume to be defined on 0, ε . By construction Φ̃_{t} F_{t}^{}Φ_{t} and Φ̃_{t}^{}F_{t}^{}Φ_{t}^{}are solutions of the gauged spinor flow equation associated with ḡ. Uniqueness of the gauged spinor flow implies that Φ̃_{t} Φ̃_{t}^{}for all t 0, ε . Hence f_{t} and f_{t}^{}are solutions of the same ODE (21). By uniqueness of the solution to ordinary differential equations, we conclude that f_{t} f_{t}^{}, whence Φ_{t} Φ_{t}^{}for all t 0, ε .

5.3. Premoduli spaces. We now study the zero locus of Q̃ḡ at a critical point ḡ, φ̄ where ḡ is an irreducible metric associated with a torsionfree G-structure. We assume n dim M 4. We claim that

Q̃ 1 ; ḡ 0 ker Xḡ Q^{1} 0 .

Indeed, assume that Qḡ Φ Q Φ λ_{Φ}^{}Xḡ Φ 0, the other inclusion being trivial. From the Bianchi identity in Corollary 4.7 we deduce λ_{Φ}Q Φ λ_{Φ}λ_{Φ}^{}Xḡ Φ λ_{Φ}λ_{Φ}^{}Xḡ Φ 0. As a consequence, λ_{Φ}λ_{Φ}^{}Xḡ Φ , Xḡ Φ _{g} 0 and therefore λ_{Φ}^{}Xḡ Φ 0 using integration by parts. This in turn implies Q Φ 0 and that X is Killing for g. Since ḡ is irreducible the fundamental group of M must be finite (cf. the arguments of Lemma 4.20), hence the isometry group of g must be finite for g is Ricci-flat. It follows that Xḡ g 0 as claimed. Now Q^{1} 0 is a smooth manifold near Φ̄ by Theorem 4.17. Further, the intersection with ker Xḡ is transversal. Indeed, Xḡ g 2 δḡg ^{}0 implies that g lies in the Ebin slice [15] of divergence-free metrics. Since Q̃ḡ is elliptic, Q̃ 1 ; ḡ 0 is smooth near g, φ . Strictly speaking we would need to pass to some suitable completion of our function spaces, but we gloss over these technicalities in view of the ellipticity of Q̃ḡ. The tangent space is given by

TΦ̄Q̃ 1 ; ḡ 0 ker Xḡ ker LΦ̄ ġ, φ̇ δ_{g}ġ 0, ^{g}φ̇ 0, _{g}Ψ_{ġ,φ} 0 ,

for ġ ker Dḡscal, the kernel of the linearisation g scal^{g}. If ḡ is Ricci-flat and if ġ is divergence-free, then Tr_{g}ġ const [8, Theorem 1.174 (e)], and the claim follows from [40, Proposition 2.2]. In fact, we can identify TΦ̄Q̃ 1 ; ḡ 0 with the second cohomology group of the elliptic complex (compare also with the cohomological interpretation of [40, Proposition 2.13]). In summary, we have proven

Theorem 5.13. Under the assumptions of Theorem 4.17, Q̃ 1 ; ḡ 0 is a slice for the { \atop Diff}_{0} M -action on Crit . In particular, the premoduli space of parallel spinors Crit { \atop Diff}_{0} M is smooth at ḡ, φ̄ .

6. The s-energy functional

In this section we study a family of spinorial energy functionals, depending on a parameter s ℝ, which generalise the spinorial energy functional. In the sevendimensional real case these functionals are related to the (generalised) Dirichlet functional(s) of the second and third author [42], [43].

Definition 6.1. For s ℝ the s-energy functional _{s} is defined by

s Φ Φ s g_{Φ} ,

where g _{M} scal^{g} dv^{g} is the total scalar curvature functional.

Remark 6.2. (i) If dim M 2 then by the Gauß-Bonnet theorem the functionals s differ from _{0} only by a constant. In the following we therefore assume that dim M 3.

(ii) As the functional _{s} and only differ by a multiple of , it is evident that all symmetries of discussed in Section 4.1 are symmetries of _{s} as well.

Example. Let D_{g} Γ Σ_{g}M Γ Σ_{g}M be the Dirac operator. Then using the Weitzenböck formula D_{g}^{2}φ ^{g} ^{g}φ \frac{1}{4}scal^{g}φ we obtain

\frac{1}{2} _{M} D_{g}φ ^{2} dv^{g} \frac{1}{2} _{M} D_{g}^{2}φ, φ dv^{g} g, φ \frac{1}{8} g ,

that is

\frac{1}{8} Φ \frac{1}{2} _{M} D_{g}φ ^{2} dv^{g}.

6.1. Short-time existence and uniqueness. In this subsection we determine the closed interval s_{min}, s_{max} of values of s for which the negative gradient flow of the s-energy functional _{s} is again weakly parabolic. In particular, 0 s_{min}, s_{max} . For values of s in the open interval s_{min}, s_{max} we will show that adding the same diffeomorphism term as before yields a strongly parabolic flow equation, providing a proof of short-time existence and uniqueness for the “s-energy flow”.

Definition 6.3. Let Q_{s} grad _{s} Q s grad . We call a solution to

\frac{∂}{∂t}Φ_{t} Q_{s} Φ_{t} , Φ_{0} Φ  (23)

the s-energy flow with initial condition Φ g, φ or s-energy flow for short.

The first variation of the total scalar curvature g _{M} scal^{g} dv^{g} is well-known. Recall that one has

\frac{d}{dt} _{t 0} g th D_{g} h _{M} Ric^{g} \frac{1}{2}scal^{g} g, h _{g} dv^{g},

see for example [8, Proposition 4.17], whence

grad g Ric^{g} \frac{1}{2}scal^{g} g.

Now

D_{g}Ric h \frac{1}{2}Δ_{L}h δ_{g}^{}δ_{g}h \frac{1}{2} ^{g}d Tr_{g}h

where Δ_{L}h ^{g} ^{g}h Ric^{g} h h Ric^{g} 2R̊^{g}h is the Lichnerowicz Laplacian mentioned in Section 4.3, and

D_{g}scal h Δ_{g}Tr_{g}h δ_{g} δ_{g}h Ric^{g}, h _{g},

see for example in [8, Theorem 1.174]. It follows that

D_{g}grad h D_{g}Ric h \frac{1}{2} D_{g}scal h
\frac{1}{2}Δ_{L}h δ_{g}^{}δ_{g}h \frac{1}{2} ^{g}d Tr_{g}h \frac{1}{2}Δ_{g} Tr_{g}h g \frac{1}{2}δ_{g} δ_{g}h g

\frac{1}{2} Ric^{g}, h _{g} g \frac{1}{2}scal^{g} h

\frac{1}{2} ^{g} ^{g}h δ_{g}^{}δ_{g}h \frac{1}{2} ^{g}d Tr_{g}h \frac{1}{2}Δ_{g} Tr_{g}h g \frac{1}{2}δ_{g} δ_{g}h g

terms of lower order in h

and hence for ξ T_{x}^{}M

σ_{ξ} D_{g}grad ġ_{x}  \frac{1}{2} ξ ^{2}ġ_{x} ξ ġ_{x} ξ,  \frac{1}{2} Tr_{g}ġ_{x} ξ ξ
\frac{1}{2} ξ ^{2} Tr_{g}ġ_{x} g \frac{1}{2} ġ_{x} ξ, ξ g.  (24)

Lemma 6.4. Let ξ T_{x}^{}M with ξ 1. Then with respect to the decomposition ġ_{x} aξ ξ ξ α α ξ γ, where a ℝ, g α, ξ 0 and γ ξ, 0 (see the proof of Corollary 5.6), one has

σ_{ξ} D_{g}grad ġ_{x} \frac{1}{2}γ \frac{1}{2} Tr_{g}γ g ξ ξ .

In particular,

g σ_{ξ} D_{g}grad ġ_{x} , ġ_{x} \frac{1}{2} γ ^{2} \frac{1}{2} Tr_{g}γ ^{2}.

Proof. Using equation (24) one easily computes

σ_{ξ} D_{g}grad ξ ξ \frac{1}{2} 1 \frac{1}{2} ξ ξ \frac{1}{2} \frac{1}{2} g 0

and

σ_{ξ} D_{g}grad ξ α \frac{1}{2} \frac{1}{2} ξ α 0.

Furthermore,

σ_{ξ} D_{g}grad γ \frac{1}{2}γ \frac{1}{2} Tr_{g}γ ξ ξ \frac{1}{2} Tr_{g}γ g,

which proves the first claim. The second one follows from g γ, g ξ ξ Tr_{g}γ. □

We consider Q_{s} Q s grad as before and put

{ \atop Q_{ḡ,s}} g, φ Q_{s} g, φ λ_{g,φ}^{}Xḡ g

with the same vector field Xḡ g 2 δḡg ^{}as in Section 5. As in the case of the ordinary spinor flow we consider the gauged version of the s-energy flow equation

\frac{∂}{∂t} Φ̃_{t} { \atop Q_{ḡ,s}} Φ̃_{t} , Φ̃_{0} Φ̄

with initial condition Φ̄ ḡ, φ̄ .

Corollary 6.5. Let n dim M 3.

(i) Q_{s} is weakly elliptic at Φ if and only if s \frac{1}{8 n 2} , \frac{1}{8} .

(ii) { \atop Q_{ḡ,s}} is strongly elliptic at Φ̄ ḡ, φ̄ if and only if s \frac{1}{8 n 2} , \frac{1}{8} .

Proof. By definition one has D_{Φ}Q_{s} D_{Φ}Q sD_{g}grad for Φ g, φ . Hence

g σ_{ξ} D_{Φ}Q_{s} ġ_{x}, φ̇_{x} , ġ_{x}, φ̇_{x} g σ_{ξ} D_{Φ}Q ġ_{x}, φ̇_{x} , ġ_{x}, φ̇_{x}
sg σ_{ξ} D_{g}grad ġ_{x} , ġ_{x} .

Now, using the notation of Corollary 5.6

\frac{1}{16} γ ^{2} s \frac{1}{2} γ ^{2} \frac{1}{2} Tr_{g}γ ^{2}   ;  ;  ;  ;  ;  ; \frac{1}{16} \frac{s}{2} γ ^{2}  Tr_{g}γ 0 \frac{1}{16} s \frac{n 2}{2} γ ^{2}  γ pure trace

from which the assertions follow using the previous calculations for the individual symbols. □

Remark 6.6. Note that \frac{1}{16} _{1 16} and s \frac{1}{16} lies in the range where { \atop Q_{ḡ,s}} is strongly elliptic. In particular, using results from Subsection 6.2 we recover Theorem 5.1 of [42]. On the other hand, as observed earlier, \frac{1}{2} _{M} D_{g}φ ^{2}dv^{g} _{1 8} and s \frac{1}{8} s_{max} is the borderline case where Q_{s} is still weakly elliptic, but { \atop Q_{ḡ,s}} is not strongly elliptic. The reason for this is that minimisers of _{1 8} are harmonic spinors. This is a far less restrictive condition on a metric than to admit parallel spinors. In particular, we get an infinite dimensional moduli space of critical points as the following example shows.

Example. Let M, g_{0} be a K3-surface with a Ricci-flat Kähler metric. The spinor bundle of a 4-dimensional manifold carries a quaternionic structure, and the space of parallel spinors on M, g_{0} is a quaternionic vector space of dimension 1. Let g_{1} be a non-Ricci-flat metric on M. Then there are no parallel spinors on M, g_{1} . We choose g_{1} is C^{1}-close to g_{0}. Due to the index theorem, the space of harmonic spinors on M, g_{1} has quaternionic dimension 1. Further, any non-trivial harmonic spinor has no zero. Finally, we also assume that g_{1} is chosen such that there are no non-trivial conformal maps from M, g_{1} to itself. If we set g̃_{1} φ ^{4 n 1} g_{1} for a harmonic spinor φ, then every harmonic spinor on M, g̃_{1} has constant length, the group of unit quaternions (or equivalently SU 2 ) acts freely and transitively on the harmonic unit spinors over g̃_{1}, and up to scaling g̃_{1} is the only metric in g_{1} with harmonic unit spinors. Thus close to g_{1} , Crit _{1 8} is an ℝ^{}SU 2 -principal bundle over the space of conformal structures, and thus, close to g̃_{1}, the moduli space Crit _{1 8} { \atop Diff}_{0} M is smooth and infinite-dimensional which is not possible if { \atop Q_{ḡ,1 8}} were strongly elliptic. Similar examples exists in other dimensions.

The same proof as for the spinor flow yields

Theorem 6.7. Let n dim M 3. Then for any s \frac{1}{8 n 2} , \frac{1}{8} the s-energy flow equation (23) has a unique short-time solution.

6.2. The generalised Dirichlet energy functional. From a principal fibre bundle point of view special holonomy metrics correspond to certain orthonormal frame bundles. These arise as extensions of G-subbundles to which the Levi-Civita connection reduces. Instead of using spinors one can describe these reduced G-bundles by means of forms of special algebraic type. In particular, one can characterise G_{2}-bundles in terms of positive 3-forms, that is, global sections of the fibre bundle associated with the open cone Λ_{}Λ^{3}ℝ^{7}, the orbit of GL_{7}^{}diffeomorphic to GL_{7}^{}G_{2}. More concretely, assume that we have a pair g, φ over a seven dimensional manifold. Take an oriented g-orthonormal basis E_{1}, …, E_{7} of ℝ^{7}, g and consider Σ_{7}^{ℝ} the real spin representation of Spin 7 . The map

φ ψ Σ_{7}^{ℝ} Σ_{7}^{ℝ} \frac{1}{4} _{p} _{1 i_{1} ... i_{p} n} φ, E_{i_{1}} … E_{i_{p}} ψ E_{i_{1}} … E_{i_{p}} Λ^{}ℝ^{7} (25)

is a Spin 7 -equivariant embedding. We require the factor 1 4 to turn this map into an isometry, that is, , , g_{Λ^{}}(recall our convention from Section 2 according to which E_{i_{1}} … E_{i_{p}} has unit norm). Choosing an explicit representation such that vol_{g} φ φ gives

φ φ \frac{1}{4} 1 Ω _{g}Ω vol_{g} ,  (26)

(cf. [31, Theorem IV.10.19] modulo our conventions; also note that a representation with vol_{g} φ φ would result in different signs of the homogeneous components). By equivariance, all theses algebraic features make sense on a seven-dimensional spin manifold M^{7} so that we get a well-defined 3-form Ω which one checks to be positive. Conversely, such a 3-form gives rise to a well-defined metric g_{Ω} with unit spinor φ_{Ω} (see for instance [21]).

It follows at once that if φ is parallel (i.e. we have holonomy G_{2}), then Ω is parallel. In fact consider the twisted Dirac operator _{g} Γ Σ_{g}M Σ_{g}M Γ Σ_{g}M Σ_{g}M which is locally defined by _{g} φ φ D_{g}φ φ _{k} e_{k} φ _{e_{k}}^{g}φ. Under the map (25) _{g} corresponds to d d^{}(see for instance [31, Theorem II.5.12]). Hence Ω is closed and coclosed if and only if _{g} φ φ 0. On the other hand contracting the latter equation with , φ in the second slot we obtain

0 φ ^{2}D_{g}φ _{k} _{e_{k}}^{g}φ, φ e_{k} φ D_{g}φ

and thus _{k} e_{k} φ _{e_{k}}^{g}φ 0. Then contracting from the right with e_{l} φ, gives _{e_{l}}^{g}φ for l 1, …, 7, whence ^{g}φ 0. We thus recover a theorem of Fernández and Gray [16] which asserts that dΩ 0, d _{g}Ω 0 is equivalent to ^{g}Ω 0. In [42] and [43], the second and third author considered the functionals

Ω_{}ℝ, Ω \frac{1}{2} _{M} dΩ _{g_{Ω}}^{2} d _{g_{Ω}}Ω _{g_{Ω}}^{2} dv^{g_{Ω}},

where the metric g_{Ω} is induced by Ω, and

Ω_{}ℝ, Ω \frac{1}{2} _{M} ^{g_{Ω}}Ω _{g_{Ω}}^{2}dv^{g_{Ω}}.

These belong to the family of generalised Dirichlet energy functionals. The critical points are the absolute minimisers which correspond, as we have just seen, to holonomy G_{2}-structures. Further, the negative gradient flow exists for short times and is unique. In fact, these functionals as well as the induced flows can be regarded as a special case of the spinorial flow resp. the s-energy flow.

Proposition 6.8. Let _{ℝ} be the bundle of universal real unit spinors associated with the real Spin 7 -representation Σ_{7}^{ℝ}. Under the isomorphism (25) we have

16 _{0} _{_{ℝ}} and 16 _{1 16} _{_{ℝ}} .

Proof. We have

_{X}^{g}φ φ φ _{X}^{g}φ \frac{1}{4} _{X}^{g}Ω _{X}^{g} _{Ω}Ω

and _{X}^{g}φ, φ 0 for φ has unit length, so

2 _{X}^{g}φ ^{2}  _{X}^{g}φ φ φ _{X}^{g}φ ^{2} \frac{1}{16}  _{X}^{g}Ω ^{2}  _{X}^{g} _{Ω}Ω ^{2}  \frac{1}{8} _{X}^{g}Ω ^{2},

and 16 ^{g}φ ^{2} ^{g}Ω ^{2} by summing over an orthogonal frame. This gives the result for . Furthermore, starting again from (26) and using the correspondence d d^{} g,

d d^{}Ω _{Ω}Ω ^{2} 16 D_{g}φ φ _{k} e_{k} φ _{e_{k}}^{g}φ, D_{g}φ φ _{ℓ} e_{ℓ} φ _{e_{ℓ}}^{g}φ

16 D_{g}φ, D_{g}φ φ ^{2} _{k,ℓ} e_{k} φ, e_{ℓ} φ _{e_{k}}^{g}φ, _{e_{ℓ}}^{g}φ

16 D_{g}φ ^{2} ^{g}φ ^{2} .

Integration yields the desired assertion. □

Since by Corollary 4.4 reality of a spinor is preserved under the flow, we immediately deduce

Corollary 6.9. Under the isomorphism (25) the negative gradient flows of and correspond to certain s-energy flows.

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Fakultät für Mathematik, Universität Regensburg, Universitätsstraße 40, D–93040 Regensburg, Germany

E-mail address: bernd.ammann@mathematik.uni-regensburg.de

Mathematisches Seminar der Universität Kiel, Ludewig-Meyn Straße 4, D–24098 Kiel, Germany

E-mail address: weiss@math.uni-kiel.de

Institut für Geometrie und Topologie der Universität Stuttgart, Pfaffenwaldring 57, D–70569 Stuttgart, Germany

E-mail address: frederik.witt@mathematik.uni-stuttgart.de