Lower bounds for eigenvalues of the Dirac operator on n-spheres with SO(n)-symmetry

Margarita Kraus

Abstract

In this paper we derive estimates for the eigenvalues of the Dirac operator and their multiplicity on manifolds diffeomorphic to S^{n} with an isometric SO(n)-action. Especially we prove a new lower bound for the first eigenvalue and show an example, where this new bound coincides in the limit with the known upper bounds. Mathematics Subject Classification: 53C25, 58G25, 58G35 Keywords: Dirac operator, spectrum, sphere, SO(n)-action

1 Introduction

The Dirac operator on a Riemannian spin manifold M is a natural first order elliptic differential operator. For closed M its spectrum is discrete and real.

There are a few examples, where the spectrum is explicitly computed. For homogeneous spaces the calculation can be reduced to representation theoretic computations, which have been done in several examples. For the spheres they result in the following theorem:

Theorem 1 ([22]) On the sphere S^{n} of constant sectional curvature 1 the Dirac operator has the eigenvalues

±(\frac{n}{2} + k), k ≥ 0  (1)

with multiplicity 2^{[ \frac{n}{2} ]}\binom{k+n−1}{k}.

More examples, where the spectrum of the Dirac operator is well known are listed in [2], e.g. the flat tori ([14]), complex projective spaces ([11], [12], [21]) and spherical space forms ([6]).

In general the Dirac operator has a nontrivial kernel (cf. [17], examples cf. [14] and [7]), but for n-dimensional manifolds M^{n} with positive scalar curvature R one deduces from the Lichnerowicz-formula that the kernel is trivial. In this case Th. Friedrich has proved in [13] that the eigenvalues satisfy

λ^{2} ≥ \frac{1}{4} \frac{n}{n − 1}R_{min}  (2)

with R_{min} := min{R(m)|m ∈ M^{n}}.

For 2-dimensional manifolds (M^{2}, g) of genus zero the Dirac operator has no kernel, because any metric g on S^{2} is conformally equivalent to the standard metric and in [5] and [20] further lower bounds for the first eigenvalue have been shown. If (M^{2}, g) with M^{2} ≅ S^{2} has additionally an S^{1}-symmetry, then in [18] we proved that the first eigenvalue satisfies

λ ≥ \frac{1}{2f_{max}}  (3)

where 2πf_{max} is the maximal volume of S^{1}-orbits. This result will be generalised in this paper:

Theorem 2 Let (M^{n+1}, g) be a Riemannian manifold, M^{n+1} ≅ S^{n+1} with an isometric SO(n + 1)-action. Let f_{max} · vol(S^{n}) be the maximal volume of orbits. Then the spectrum of M is symmetric. The multiplicity of an eigenvalue λ with

\frac{\frac{n}{2} + k}{f_{max}} ≤ λ < \frac{\frac{n}{2} + k + 1}{f_{max}} (k ≥ 0)

is at most 2^{[ \frac{n+1}{2} ]}\binom{n+k}{k}.

Especially the smallest positive eigenvalue satisfies λ ≥ \frac{n}{2f_{max}} and has multiplicity at most 2^{[ \frac{n+1}{2} ]}.

There are some well known upper bounds for the first positive eigenvalue of the Dirac operator, e.g. for compact connected Riemannian spin manifolds of dimension 2m and positive sectional curvature K with maximum K_{max} the inequality

λ^{2} ≤ 4^{m−1}mK_{max}

for the first positive eigenvalue holds ([4]). More upper bounds are proved in dimension 2 (cf. [1]) and for manifolds M^{n}, which are isometrically immersed in IR^{n+1} (e.g. [4], [10], [16]). We will use the following inequality for the first eigenvalue to show that the new lower bound is sharp in the limit.

Lemma 1 ([8], Corollary 4.2) Let M be an n-dimensional closed oriented hypersurface isometrically immersed in IR^{n+1}. Let M carry the induced spin structure. Let H be the mean curvature of M in IR^{n+1}.

Then the classical Dirac operator D_{M} of M has at least 2^{[ \frac{n}{2} ]} eigenvalues λ (counted with multiplicities) satisfying

λ^{2} ≤ \frac{n^{2} ∫ H^{2}dM}{4vol(M)}  (4)

We will show an example where the new lower bound coincides in the limit with this upper bound (4) even with multiplicity.

In the next section we recall some facts about Dirac operators on hypersurfaces and warped products and apply them to the space of principal orbits. This reduces the Dirac equation on the SO(n + 1)-manifold S^{n+1} to a system of first order differential equations. In the third section this will be used to prove theorem 2.

2 The Dirac operator on the bundle of principal orbits

Let (M, g) be an n+1-dimensional oriented Riemannian manifold M ≅ S^{n+1} with an isometric effective SO(n + 1)-action. Then the SO(n + 1)-action has only principal orbits which are spheres S^{n} and two fixed points ([9], p. 185 and p. 230). We denote with M_{0} the dense submanifold of principal orbits.

Let γ : I = (0, L) → M_{0} be a section of M_{0} → M_{0}/SO(n + 1) orthogonal to the fibres parametrised by arc length and

f : I → IR^{+}, f(t) = \frac{vol(SO(n + 1) · γ(t))}{2π^{\frac{n+1}{2}}}Γ(\frac{n + 1}{2})

then

(M_{0}, g) = (I × S^{n}, dt^{2} + f^{2}(t)dϕ^{2}) =: I ×_{f} S^{n}

where dϕ^{2} denotes the standard metric on S^{n}. For given f the closure of I ×_{f} S^{n} is a C^{∞}-manifold iff ḟ(0) = −ḟ(L) = 1 and f extends at zero to an odd smooth function and similarly at L.

To calculate the Dirac operator on the space of principal orbits M_{0}, we use some facts about spinor bundles on hypersurfaces and warped products which have been developed in [3] and similarly in [7], pp 902, with a slight difference in the case n odd. We will follow [7] in this paper. For details concerning the Dirac operator references are [19] and [15].

Let X^{n+1} be an (n+1)-dimensional spin manifold with spin structure Q_{X} and Y^{n} ⊂ X^{n+1} an n-dimensional closed submanifold with the induced spin structure Q_{Y}, such that the restriction of the spinor bundle over X to Y is given by

Σ_{X}|_{Y} = Q_{Y} ×_{Spin(n)} Δ_{n+1}

where Δ_{n+1} denotes the (n + 1)-dimensional spinor-representation.

If n is odd Δ_{n+1} admits a splitting Δ_{n+1} = Δ_{n+1}^{+} ⊕ Δ_{n+1}^{−} into two irreducible spin representations. The canonical algebra isomorphism Cl(IR^{n}) → Cl^{0}(IR^{n+1}) of the Clifford algebra of IR^{n} in the even part of the Clifford algebra of IR^{n+1} induced by e_{i} ↦ e_{i} · e_{n+1} for the standardbasis (e_{1}, …, e_{n+1}) of IR^{n+1} makes Δ_{n+1} into two representations of Cl(IR^{n}). Therefore the following relation of the spinor bundles Σ_{X} and Σ_{Y} over X and Y holds:

Σ_{X}|_{Y} = Σ_{X}^{+}|_{Y} ⊕ Σ_{X}^{−}|_{Y} = Σ_{Y} ⊕ Σ̂_{Y}

The induced Clifford multiplication with vectors v ∈ T Y on Σ_{Y} and Σ̂_{Y} differs in the sign and Clifford multiplication with a normal unit vectorfield ν on Y gives an isomorphism Σ_{Y} → Σ̂_{Y} which anticommutes with the Clifford multiplication.

For n even the algebra isomorphism Cl(IR^{n}) → Cl^{0}(IR^{n+1}) makes Δ_{n+1} into a representation of Cl(IR^{n}) which is the usual spinor representation of Cl(IR^{n}). Therefore we get

Σ_{X}|_{Y} ≅ Σ_{Y} = Σ_{Y}^{+} ⊕ Σ_{Y}^{−}.

Let ψ be a spinor on X defined on a neighbourhood of Y then the classical Gauss-formula for the Levi-Civita connection gives for the Dirac operator

−ν · D^{X}ψ = D^{Y} ψ − \frac{n}{2}Hψ + ∇_{ν}ψ

where D^{X} is the Dirac operator on Σ_{X} and D^{Y} is the Dirac operator on Σ_{Y} if n is even and the sum of the Dirac operator and its negative if n is odd, and H is the mean curvature of Y in X. A straightforward calculation shows that the Dirac operator D^{Y} anticommutes with the Clifford multiplication with a normal unit vectorfield ν.

Now let M_{0} = I ×_{f} S^{n}, with I = [0, L] and f : I → IR^{+}. We denote with Σ_{M_{0}} the spinor bundle over M_{0} and with Σ the spinor bundle over S^{n}. Then

Σ_{M_{0}} ≅ π^{∗}Σ ⊕ π^{∗}Σ̂ for n odd  (5)
Σ_{M_{0}} ≅ π^{∗}Σ  for n even  (6)

with π : I ×_{f} S^{n} → S^{n}.

Let {σ_{j} ∈ ΓΣ}_{j∈Z Z\{0}} be an L^{2}-orthonormalbasis of eigenspinors in Σ with Dσ_{j} = k_{j}σ_{j}. We choose the numbering of σ_{j} such that k_{j} > 0 for j > 0 and k_{−j} = −k_{j}

Because of D^{N}(ν · ψ) = −ν · D^{N}ψ we can choose for n even σ_{−j} := ν · σ_{j}. For n odd let σ̂_{−j} := ν · σ_{j} be a basis of eigenspinors in Σ̂. Now we denote with σ_{j} and σ̂_{j} also the sections in Σ_{M_{0}} given by the isomorphism (5) and (6).

Then a smooth spinor ψ ∈ ΓΣ_{M_{0}} can be expressed as

ψ = Σ_{j∈Z Z\{0}}θ_{j}σ_{j} + Σ_{j∈Z Z\{0}}θ̂_{j}σ̂_{j} for n odd ,  (7)
ψ = Σ_{j∈Z Z\{0}}θ_{j}σ_{j}  for n even  (8)

with θ_{j}, θ̂_{j} ∈ C^{∞}([0, L], C).

Lemma 2 ([7], p. 919) A spinor ψ = Σ_{j∈Z Z\{0}}θ_{j}σ_{j} ∈ ΓΣ_{M_{0}} respectively ψ = Σ_{j∈Z Z\{0}}θ_{j}σ_{j} + Σ_{j∈Z Z\{0}}θ̂_{j}σ̂_{j} ∈ ΓΣ_{M_{0}} satisfies Dψ = λψ iff θ_{j} and θ̂_{j} satisfy the equations

θ̇_{j} = [−\frac{n}{2} \frac{ḟ}{f} − \frac{k_{j}}{f}]θ_{j} + λθ_{−j}  (9)
θ̇_{−j} = [−\frac{n}{2} \frac{ḟ}{f} + \frac{k_{j}}{f}]θ_{−j} − λθ_{j}  (10)

for all j ∈ IN for n even and the same equations with θ_{−j} replaced by θ̂_{−j} for all j ∈ ZZ\{0} for n odd.

3 Lower bounds for the eigenvalues and multiplicities

To study the eigenvalues of the Dirac operator on an SO(n + 1)-manifold M ≅ S^{n+1} with M_{0} = I ×_{f} S^{n} it is sufficient to restrict to analytic functions f because a small deformation of the metric in the C^{1}-topology does not change the eigenvalues of the Dirac operator too much, more exactly:

Lemma 3 ([7], prop. 7.1) For ϵ > 0, Λ > 0 there exists a C^{1}-neighbourhood of g, such that for any g̃ in this neighbourhood with associated Dirac operator D̃ and any λ ∈ [−Λ, Λ] we have

dim E_{{λ}}(D) ≤ dim E_{[λ−ϵ,λ+ϵ]}(D̃) ≤ dim E_{[λ−2ϵ,λ+2ϵ]}(D)

where E_{[a,b]} is the direct sum of eigenspaces of D for the eigenvalues λ ∈ [a, b].

Equations (9) and (10) have been studied in [18]. The restriction of a smooth spinor on M to M_{0} = I ×_{f} S^{n} gives a spinor whose coefficient functions θ_{j} and θ̂_{j} in the decomposition (7) and (8) are bounded. The following Lemma generalizes a result of [18].

Lemma 4 If θ_{j} and θ_{−j} are solutions of (9) and (10), k_{j} > 0 and θ_{j} and θ_{−j} are bounded at 0 and L, then |\frac{θ_{−j}(0)}{θ_{j}(0)}| = ∞ and |\frac{θ_{−j}(L)}{θ_{j}(L)}| = 0.

Proof Solutions θ_{j} and θ_{−j} of (9) and (10) are solutions of

θ̈_{j} = −n\frac{ḟ}{f}θ̇_{j} + (\frac{(2k_{j} + ḟ)^{2} − (n − 1)^{2}ḟ^{2}}{4f^{2}} − \frac{n}{2} \frac{f̈}{f} − λ^{2})θ_{j}
θ̈_{−j} = −n\frac{ḟ}{f}θ̇_{−j} + (\frac{(2k_{j} − ḟ)^{2} − (n − 1)^{2}ḟ^{2}}{4f^{2}} − \frac{n}{2} \frac{f̈}{f} − λ^{2})θ_{−j}

which are regular singular at 0 and L with characteristic indices at 0:

ν_{j,1} = k_{j} − \frac{n}{2} + 1 ν_{j,2} = −k_{j} − \frac{n}{2}
ν_{−j,1} = k_{j} − \frac{n}{2}  ν_{−j,2} = −k_{j} − \frac{n}{2} + 1

and at L:

ν_{j,1} = k_{j} − \frac{n}{2}  ν_{j,2} = −k_{j} − \frac{n}{2} + 1
ν_{−j,1} = k_{j} − \frac{n}{2} + 1 ν_{−j,2} = −k_{j} − \frac{n}{2}

If θ_{j} and θ_{−j} are solutions of (9) and (10) which are bounded at 0 and L, then \frac{−θ_{−j}}{θ_{j}} is a (possibly singular) solution of the real Riccati equation

ż = λz^{2} + \frac{2k_{j}}{f} z + λ  (11)

with boundary conditions

|z(0)| = ∞ z(L) = 0  (12)

Lemma 5 ([18], Lemma 3) If (11) has a solution with boundary conditions (12) for k_{j} > 0 and λ > 0, then

λ ≥ \frac{k_{j}}{f_{max}}  (13)

with f_{max} := max_{t∈[0,L]}f(t).

This yields for the Dirac operator:

Theorem 2 Let M^{n+1} ≅ S^{n+1} be a manifold with an effective isometric SO(n + 1)-action. Let M_{0} = I ×_{f} S^{n} be the space of principal orbits. Then the spectrum of M is symmetric. The multiplicity of an eigenvalue λ with

\frac{\frac{n}{2} + k}{f_{max}} ≤ λ < \frac{\frac{n}{2} + k + 1}{f_{max}} (k ≥ 0)

is at most 2^{[ \frac{n+1}{2} ]}\binom{n+k}{k}. Especially the smallest positive eigenvalue satisfies λ ≥ \frac{n}{2f_{max}} and has multiplicity at most 2^{[ \frac{n+1}{2} ]}. Proof If θ_{j}σ_{j} +θ_{−j}σ_{−j} for n even respectively θ_{j}σ_{j} +θ̂_{−j}σ̂_{−j} for n odd is an eigenspinor for the eigenvalue λ, then θ_{j}σ_{j} −θ_{−j}σ_{−j} respectively θ_{j}σ_{j} −θ̂_{−j}σ̂_{−j} is an eigenspinor for the eigenvalue −λ.

For k_{j} = \frac{n}{2} + k according to (13) the system of equations (9) and (10) has solutions only for λ ≥ \frac{\frac{n}{2} +k}{f_{max}}.

For analytic f the space of bounded solutions of (9) and (10) is at most 1-dimensional, because the system is weakly singular at 0 and L with characteristic indices k_{j} − \frac{n}{2} and −k_{j} − \frac{n}{2} at 0. Further on the multiplicity of an eigenvalue k_{j} = \frac{n}{2} + j over S^{n} is 2^{[ \frac{n}{2} ]}\binom{n+j−1}{j}.

Therefore for n even the multiplicity of an eigenvalue

\frac{\frac{n}{2} + k}{f_{max}} ≤ λ < \frac{\frac{n}{2} + k + 1}{f_{max}}

is at most

Σ_{l=0}^{k}2^{\frac{n}{2}} \binom{n + l − 1}{l} = 2^{\frac{n}{2}} \binom{n + k}{k}.

For n odd in equation (9) and (10) also j < 0 occur. If θ_{j}σ_{j} + θ̂_{−j}σ̂_{−j} is an eigenspinor with eigenvalue λ than also θ̂_{−j}σ_{−j} − θ_{j}σ̂_{j} because (θ̂_{−j}, −θ_{j}) solves the system (9) and (10) with k_{−j} = −k_{j}. Therefore the multiplicity is at most 2^{\frac{n+1}{2}} \binom{n+k}{k}.

If M is an (n + 1)-dimensional sphere, which means M_{0} = I ×_{sin} S^{n}, then the well known multiplicities of the eigenvalues are exactly the maximal multiplicities according to the theorem.

Finally we show that the lower bound of theorem 2 coincides in the limit with the upper bound (4):

Consider the cylinder [0, L]×S^{n} with hemispheres glued to it at the ends. In this example the estimate (2) gives no lower bound. According to (4) there are at least 2^{[ \frac{n+1}{2} ]} eigenspinors for the eigenvalue

λ^{2}(L) ≤ \frac{n^{2}Lvol(S^{n}) + (n + 1)^{2}vol(S^{n+1})}{4(Lvol(S^{n}) + vol(S^{n+1}))}

For L → ∞ this gives

\overline{lim}_{L→∞}λ(L) ≤ \frac{n}{2}.

On the other hand the new lower bound yields

λ(L) ≥ \frac{n}{2} for every L

with multiplicity at most 2^{[ \frac{n+1}{2} ]}.

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