Lower bounds for eigenvalues of the Dirac operator on surfaces of rotation

Margarita Kraus

Abstract

In this paper we will prove new lower bounds for the first eigenvalue of the Dirac operator on 2-dimensional Riemannian manifolds diffeomorphic to S^{2} with an isometric S^{1}-action. We show examples, where this new bound improves the known lower bounds and coincides in the limit with the known upper bounds. Mathematics Subject Classification: 53C25, 58G25, 58G30 Keywords: Dirac operator, spectrum, surface of rotation

1 Introduction

On closed Riemannian spin manifolds the spectrum of the Dirac operator is discrete and real. There are a few examples where the spectrum is well known, e.g. the flat tori ([12]), spheres of constant curvature ([17]) and spherical space forms ([4]) (for an overview and more examples cf. [2]). For the sphere S^{n} of constant sectional curvature 1 the Dirac operator has the eigenvalues

±(\frac{n}{2} + k), k ≥ 0

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

In general there are only estimates for the eigenvalues of the Dirac operator. For an n-dimensional closed Riemannian spin manifold M^{n} with scalar curvature R, Th. Friedrich has proved in [13] that the eigenvalues satisfy

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

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 the inequality

λ^{2} ≥ \frac{4π}{vol(S^{2}, g)}  (2)

holds for the eigenvalues (cf. [16], [5], [9]).

There are also some upper bounds for the first positive eigenvalue of the Dirac operator, e.g. in [8] it is shown that 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.

An other intrinsic upper bound for Riemannian manifolds M^{2} diffeomorphic to S^{2} is given in [1]: Let

δ_{c}^{Dirac}(M^{2}, g) = inf\{∫_{S^{2}} \frac{|grad(h_{Φ})|^{2}}{h_{Φ}^{2}} dS^{2} : Φ ∈ U(S^{2}, M^{2})\}

where U(S^{2}, M^{2}) denotes the set of all uniformisation maps preserving the orientation. Then

λ_{1}^{2} ≤ \frac{4π}{vol(M^{2}, g)} + \frac{δ_{c}^{Dirac}(M^{2}, g)}{vol(M^{2}, g)}  (3)

For manifolds M^{n}, which are isometrically immersed in IR^{n+1}, extrinsic upper bounds are known (e.g. [8], [10]). In [6] and [14] the following inequality for the first positive eigenvalue of the Dirac operator on M^{n} ↪ IR^{n+1} with the induced spin structure on M^{n} is shown:

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

where H denotes the mean curvature.

In [1] the inequality

λ^{2} ≤ \frac{∫_{M^{2}} H^{2}(f^{2} + G^{2}(f))dM^{2} + ∫_{M^{2}} |gradf|^{2}(1 + [G′(f)]^{2})dM^{2}}{∫_{M^{2}}(f^{2} + G^{2}(f))dM^{2}} (5)

where f : M^{2} → IR, G : IR → IR are smooth functions and G′ denotes the derivative of G, has been proved.

In this paper we give new lower bounds for the eigenvalues of the Dirac operator on 2-dimensional manifolds of genus 0 with isometric S^{1}-action. We will show the following

Theorem Let (M^{2}, g) be a closed Riemannian manifold of genus 0 with an isometric S^{1}-action with maximum length of orbits 2πf_{max}. Then

1. λ ≥ \frac{1}{2f_{max}}

2. The multiplicity of an eigenvalue λ_{n} > 0 with

\frac{2n − 1}{2f_{max}} ≤ λ_{n} ≤ \frac{2n + 1}{2f_{max}}

is at most 2n.

On the one hand there are many examples, where this estimate improves the known lower bounds for the Dirac operator on surfaces, and on the other hand it shows that the known upper bounds and the new lower bound coincide in the limit for the ellipsoid. More exactly:

For the ellipsoid E(a) with axes (a, 1, 1) with a > 1 the minimal scalar curvature is \frac{1}{a^{2}} and the volume is 2π + \frac{4πa^{2}}{\sqrt{a^{2}−1}} arcsin \sqrt{\frac{a^{2}−1}{a^{2}}}. Therefore the estimates (1) and (2) for the smallest positive eigenvalue λ(a) on the ellipsoid E(a) give no lower bound in the limit a → ∞, whereas the theorem above shows:

λ(a) ≥ \frac{1}{2} for every a > 0.

On the other hand the inequalities (4) and (5) give

\overline{lim_{a→∞}}λ(a) ≤ \frac{1}{2}

(cf. [1]), which is the optimum result.

In the second section we will give some general facts about the Dirac operator on surfaces and calculate the eigenvalue equation in special coordinates, in the third section we will use this equation to get the estimate.

Acknowledgements I wish to thank Dr. St. Bechtluft-Sachs, Prof. Th. Friedrich and Prof. K. Jänich for helpful discussions.

2 The Dirac operator on surfaces of rotation

Let (M, g) be a 2-dimensional oriented Riemannian manifold M ≅ S^{2} with an isometric S^{1}-action, whose principal orbits are free. Then M/S^{1} ≅ [0, 1], the dense submanifold M_{0} of principal orbits is an S^{1}-fibre bundle over M_{0}/S^{1} ≅ (0, 1) (cf. [3], prop. 8.2) and the two orbits, which are not free are fixed points.

Let γ : I = (0, L) → M_{0} be a section of M_{0} → M_{0}/S^{1} orthogonal to the fibres parametrised by arc length and f : I → IR^{+}, f(t) = \frac{vol(S^{1}·γ(t))}{2π}, then

(M_{0}, g) = (I × S^{1}, dt^{2} + f^{2}(t)dϕ^{2}) =: I ×_{f} S^{1}

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

Example For the ellipsoid with axes (a, b, b) we have f(t) = b sin x(t), with t(x) = ∫_{0}^{x} \sqrt{a^{2} sin^{2} ϕ + b^{2} cos^{2} ϕ}dϕ, t ∈ (0, L), L := ∫_{0}^{π} \sqrt{a^{2} sin^{2} ϕ + b^{2} cos^{2} ϕ}dϕ.

The frame-bundle P of I ×_{f} S^{1} is trivialised by the canonical section s := (∂_{t}, \frac{∂_{ϕ}}{|∂_{ϕ}|}). S^{2} has exactly one spin structure given by the Hopf bundle. Therefore the spin bundle of M is given by

Q|I ×_{f} S^{1} ≅ I × S^{1} × S^{1} → P, (t, z_{1}, z_{2}) ↦ (t, z_{1}, z_{1}z_{2}^{2})

and the identifications (0, ϕ, z) ∼ (0, 0, z) and (1, ϕ, z) ∼ (1, 0, e^{−iϕ}z) at the poles.

The section s does not lift to a global section of Q|I ×_{f} S^{1}. A local lift s̃ of s satisfies s̃(t, 0) = −s̃(t, 2π) and the section s̃e^{iϕ/2} is globally defined on I ×_{f} S^{1}. In a neighbourhood of the poles s̃e^{iϕ/2} and s̃e^{−iϕ/2} respectively are smooth local sections of the spin-bundle over M.

The Levi-Civita connection ∇ on I ×_{f} S^{1} is given by

< ∇∂_{t}, \frac{∂_{ϕ}}{|∂_{ϕ}|} >= \frac{ḟ(t)}{f(t)}σ^{2}

with σ^{2} =< \frac{∂_{ϕ}}{|∂_{ϕ}|}, · >.

The spinor derivative for a local spinor

[s̃, ψ], with ψ =: ( \begin{matrix}ψ_{1} \\ ψ_{2} \end{matrix} ) ∈ C^{∞}(I ×_{f} S^{1}\(t, 1), C^{2})

is given by

∇ψ = dψ + \frac{i}{2} \frac{ḟ}{f} ( \begin{matrix}−ψ_{1} \\ ψ_{2} \end{matrix} ) σ^{2}

and therefore the Dirac operator by

Dψ = ( \begin{matrix}\frac{1}{f}ψ′_{2} + iψ̇_{2} + \frac{i}{2} \frac{ḟ}{f}ψ_{2} \\ −\frac{1}{f}ψ′_{1} + iψ̇_{1} + \frac{i}{2} \frac{ḟ}{f}ψ_{1} \end{matrix} )

where ψ′_{i} = ∂_{ϕ}ψ_{i} and ψ̇_{i} = ∂_{t}ψ_{i} (For details concerning the Dirac operator see [11] or [15]).

The Dirac operator preserves the Fourier decomposition of ψ, so we re-

strict to spinors of the form ψ(t, ϕ) = θ(t)e^{ikϕ} with θ =: ( \begin{matrix}θ_{1} \\ θ_{2} \end{matrix} ) ∈ C^{∞}(I, C^{2}).

Then ψ = θe^{ikϕ} defines a spinor on I ×_{f} S^{1} iff k = \frac{2n+1}{2}, n ∈ ZZ and on these spinors the Dirac operator acts as:

Dψ = i ( \begin{matrix}θ̇_{2} + \frac{1}{2f} (2k + ḟ)θ_{2} \\ θ̇_{1} + \frac{1}{2f} (−2k + ḟ)θ_{1} \end{matrix} ) e^{ikϕ}

A spinor ψ = θ(t)e^{ikϕ} ∈ C^{∞}(I ×_{f} S^{1}\(t, 1), C^{2}) is globally defined on M iff θ_{1}(t)e^{iϕ(k− \frac{1}{2} )}, θ_{1}(t − 1)e^{iϕ(k+ \frac{1}{2} )}, θ_{2}(t)e^{iϕ(k+ \frac{1}{2} )}, and θ_{2}(t − 1)e^{iϕ(k− \frac{1}{2} )} define smooth functions at 0, since s̃e^{i \frac{ϕ}{2}} is a smooth local section at 0 × S^{1} and s̃e^{−i \frac{ϕ}{2}} at 1 × S^{1}. This means that θ_{1} vanishes at zero of order |k − \frac{1}{2} | and at 1 of order |k + \frac{1}{2} | , θ_{2} vanishes at zero of order |k + \frac{1}{2} | and at 1 of order

|k − \frac{1}{2} |.

If (M, g) is a Riemannian manifold with isometric S^{1}-action, then there exists in every neighbourhood of g in the C^{1}-topology a metric g̃ such that (M, g̃) is the closure of I ×f̃ S^{1} with an analytic function f̃.

In [7] it is shown that 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 1 ([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].

So we will restrict in the following to I ×_{f} S^{1} with analytic f.

3 Lower bounds for eigenvalues

We consider the Dirac operator on an S^{1}-manifold which is the closure of I ×_{f} S^{1} with analytic f : I → IR. As shown in the previous section the eigenvalue equation for a local spinor ψ = θe^{iϕk} is given by

λθ_{1} = i(θ̇_{2} + \frac{1}{2f} (2k + ḟ)θ_{2})  (6)
λθ_{2} = i(θ̇_{1} + \frac{1}{2f} (−2k + ḟ)θ_{1})  (7)

with boundary conditions |θ_{i}(0)| < ∞, |θ_{i}(L)| < ∞. These boundary conditions already ensure that θ describes a L^{2}-spinor and thus as a solution of the eigenvalue equation a C^{∞}-spinor.

We restrict to the case λ > 0 and k > 0: If ( \begin{matrix}θ_{1}(t) \\ θ_{2}(t) \end{matrix} ) e^{iϕk} is an eigen-

spinor with eigenvalue λ, then also ( \begin{matrix}θ_{2}(t) \\ θ_{1}(t) \end{matrix} ) e^{−iϕk}, and ( \begin{matrix}θ_{1}(t) \\ −θ_{2}(t) \end{matrix} ) e^{iϕk} is an

eigenspinor with eigenvalue −λ.

Furthermore we can assume θ_{1} as imaginary and θ_{2} real.

Lemma 2 If θ_{1} and θ_{2} are solutions of (6) and (7), k > 0 and θ_{1} and θ_{2} are bounded at 0 and L, then |\frac{θ_{1}(0)}{θ_{2}(0)}| = ∞ and |\frac{θ_{1}(L)}{θ_{2}(L)}| = 0.

Proof If θ_{1}, θ_{2} are solutions of (6) and (7), then

θ̈_{1} = −\frac{ḟ}{f}θ̇_{1} + (\frac{(2k − ḟ)^{2}}{4f^{2}} − \frac{f̈}{2f} − λ^{2})θ_{1}  (8)

and

θ̈_{2} = −\frac{ḟ}{f}θ̇_{2} + (\frac{(2k + ḟ)^{2}}{4f^{2}} − \frac{f̈}{2f} − λ^{2})θ_{2}  (9)

Thus the characteristic equations of equations (8) and (9) at zero are

ν(ν − 1) + ν − \frac{1}{4}(1 − 2k)^{2} = 0
ν(ν − 1) + ν − \frac{1}{4}(1 + 2k)^{2} = 0

Therefore locally at 0 any solution of (8) and (9), which is bounded, is given by

θ_{1}(z) = C_{1}z^{k− \frac{1}{2}} (1 + h_{1}(z))
θ_{2}(z) = C_{2}z^{k+ \frac{1}{2}} (1 + h_{2}(z))

with analytic h_{i} and h_{i}(0) = 0. Indeed taking the expansion in power series in (6) we see that if θ_{1}(t) vanishes of order m at 0, then θ_{2}(t) vanishes of order m + 1. Thus |\frac{θ_{1}(0)}{θ_{2}(0)}| = ∞. The same argument at L gives |\frac{θ_{1}(L)}{θ_{2}(L)}| = 0.

Example For the sphere the equations (6) and (7) give:

λθ_{1} = i(θ̇_{2} + \frac{1}{2 sin t}(2k + cos t)θ_{2})  (10)
λθ_{2} = i(θ̇_{1} + \frac{1}{2 sin t}(−2k + cos t)θ_{1})  (11)

with the eigenfunctions θ_{1}(t) = i cos(\frac{t}{2}) and θ_{2}(t) = sin(\frac{t}{2}) for λ = 1 and k = \frac{1}{2}.

If θ_{1} and θ_{2} are solutions of (6) and (7) which are bounded at 0 and L, then i\frac{θ_{1}}{θ_{2}} is a (possibly singular) solution of the real Riccati equation

ż = λz^{2} + \frac{2k}{f} z + λ  (12)

with boundary conditions

|z(0)| = ∞ z(L) = 0  (13)

Lemma 3 If (12) has a solution with boundary conditions (13), then λ ≥ \frac{k}{f_{max}} with f_{max} := max_{t∈[0,L]}f(t).

Proof Consider the t-dependent vectorfield v_{t} on IR given by

v_{t}(z) := λz^{2} + \frac{2k}{f(t)}z + λ for t ∈ (0, L).

γ : (0, L) → IR is a solution of (12) iff γ̇(t) = v_{t}(γ(t)).

For z ≥ 0, k > 0, λ > 0 the vectorfield v_{t} satisfies v_{t}(z) > 0 for every t ∈ (0, L). This means that for every solution γ of (12) and t_{0} ∈ (0, L) with γ(t_{0}) ≥ 0 the inequality γ̇(t_{0}) > 0 holds.

If λ < \frac{k}{f_{max}} than v_{t}(−1) = 2(λ − \frac{k}{f(t)}) < 0 for every t ∈ (0, L), which means that every solution γ of (12) and t_{0} ∈ (0, L) with γ(t_{0}) = −1 satisfy γ̇(t_{0}) < 0.

Therefore for every solution γ : (0, L) → IR of (12) with γ(L) = 0 we get −1 < γ(0) < 0 if 0 < λ < \frac{k}{f_{max}}.

For the Dirac operator this yields the following result:

Theorem Let M ≅ S^{2} be a surface of rotation with M_{0} = I ×_{f} S^{1}, then

1. The smallest positive eigenvalue λ_{|k|+ \frac{1}{2}} with an eigenspinor locally given by ψ = θe^{iϕk} satisfies λ_{|k|+ \frac{1}{2}} ≥ \frac{|k|}{f_{max}}, especially the smallest eigenvalue satisfies

λ_{1} ≥ \frac{1}{2f_{max}}

2. The multiplicity of an eigenvalue λ_{|k|+ \frac{1}{2}} with \frac{|k|}{f_{max}} ≤ λ_{|k|+ \frac{1}{2}} ≤ \frac{|k|+1}{f_{max}} is at most 2|k| + 1.

Especially for the Ellipsoid with axes (a, b, b) we get

λ_{1} ≥ \frac{1}{2b} for every a > 0

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