EIGENVALUE ESTIMATES FOR THE DIRAC OPERATOR ON WARPED PRODUCTS OVER BOUNDED MANIFOLDS

MARGARITA KRAUS AND CHRISTIANE TRETTER

Abstract. In this paper we consider the Dirac operator D_{M} on manifolds M which are, apart from submanifolds of codimension k+1, warped products where the fibers are k–dimensional spheres. By means of a suitable decomposition of the space of spinors and an abstract operator theoretic result, we show for k > 1 that the spectrum of a Dirac operator D_{M} is symmetric with respect to 0 and has a gap around 0. We establish estimates for its eigenvalues, in particular we show that \frac{k}{2f_{max}} is a lower bound for the first positive eigenvalue where f_{max} is the maximum value of the warpe function f.

1. Introduction

The Dirac operator on a closed n–dimensional Riemannian spin manifold M^{n} is an essentially self-adjoint operator with discrete spectrum. For the k–sphere S^{k} its spectrum has been calculated by S. Sulanke [17]:

λ_{n,±} = ± (\frac{k}{2} + n) , n ∈ ℕ_{0}.

In general, the Dirac operator may have a kernel, but in the case of a manifold with positive scalar curvature R it has been proved by Th. Friedrich [10] that its eigenvalues satisfy the inequality

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

with R_{min} := min{R(m) : m ∈ M^{n}}. In [14] a new lower bound for the eigenvalues of the Dirac operator on manifolds diffeomorphic to S^{k+1} with an isometric effective SO(k + 1)–action has been established. E.g. the smallest positive eigenvalue λ_{0} satisfies the estimate

λ_{0} ≥ \frac{k}{2f_{max}}  (1.1)

where f_{max} · vol S^{k} is the maximal volume of orbits. Here the sphere S^{k+1} is considered without the two fixed points of the SO(k + 1)–action as a fiber bundle with fiber S^{k} over an interval I with a warped product metric.

In [1] (see also [2], [3]) B. Ammann derived an analogous result for families of fiber bundles M_{n} with fiber S^{1} over closed m–dimensional spin manifolds B. In particular, he proved that if the volume f_{n} of the fibers and the connection form iω_{n} of M_{n} → B satisfy ∥f_{n}∥_{∞} → 0, ∥f_{n}dω_{n}∥_{∞} → 0 and α := lim sup ∥gradf_{n}∥_{∞} < 1/2, then for every ε > 0 there exists an n_{0} ∈ ℕ such that for all n ≥ n_{0} the smallest eigenvalue λ_{0}(n) in modulus of the Dirac operator on M_{n} satisfies

Date: 04.09.2000.

1991 Mathematics Subject Classification. 58J50, 35P15.

The support for this work of the German Research Foundation, DFG, Grant TR 368/4–1, is gratefully acknowledged.

λ_{0}(n)^{2} ≥ \frac{\frac{1}{2}(\frac{1}{2} − α) − ε}{f_{n,max}^{2}}

for a non-projectable spin structure on M_{n} where f_{n,max} · vol S^{1} is the maximal volume of fibers of M_{n}. Here a spin structure on M_{n} is called non-projectable if the S^{1}–action on M_{n} does not lift to the spin structure.

In the present paper we will generalize the result of [14] for k > 1 by a completely different technique to Dirac operators on warped products with fiber S^{k} over m– dimensional manifolds B with boundary ∂B for arbitrary k and m. More exactly, the warpe function f : B → ℝ_{0}^{+} is a C^{∞}–function with f|_{B_{0}} > 0, f|_{∂B} ≡ 0, where B_{0} := B \ ∂B denotes the interior of B. The closure M = B ×_{f} S^{k} of the warped product B_{0} ×_{f} S^{k} = (B_{0} × S^{k}, m_{B_{0}} + f^{2}m_{S^{k}}) with a metric m_{B_{0}} on B_{0} and the standard metric m_{S^{k}} on S^{k} becomes an (m + k)–dimensional manifold under some additional conditions on the metric m_{B} on B and on f (see Lemma 2.2 below). We will call B×_{f} S^{k} a warped product with boundary. Warped products with boundary are manifolds with effective SO(k + 1)–action that have only two types of orbits, namely k–spheres and fixed points. For m = 1, examples are surfaces of revolution or the manifolds considered in [14] described above, for m > 1 examples are spheres with the standard metric or m–dimensional ellipsoids with SO(k + 1)–symmetry.

Our main result is an estimate for eigenvalues of weight μ (see Definition 7.1 below) of the Dirac operator D_{M} on a warped product with boundary M = B×_{f} S^{k}. In particular, we prove that the smallest positive eigenvalue of D_{M} satisfies an estimate (1.1) where now f_{max} := max{f(x) : x ∈ B}.

To this end, we decompose the Dirac operator D_{M_{0}} formally into a direct sum

D_{M_{0}} = ⨁_{μ∈Spec_{p}(D_{S^{k}})} D_{μ}.

This representation is induced by a decomposition of the space of spinors on the fibers S^{k} into the eigenspaces E(μ, D_{S^{k}}), μ ∈ Spec_{p}(D_{S^{k}}), of the Dirac operator on S^{k} which yields a decomposition of the space of spinors of M into subspaces W_{μ} of spinors of weight μ. The operators D_{μ} are the restrictions of D_{M_{0}} to the spaces W_{μ} and have a block operator matrix representation

D_{μ} = ( \begin{matrix}A_{μ} & C \\ C^{∗} & Â_{μ} \end{matrix} )  (1.2)

where A_{μ}, Â_{μ} and C are operators with certain domains and the spectra of the diagonal operators A_{μ} and Â_{μ} have a gap around 0. This structure allows to apply an abstract operator theoretic result proved in [15] which shows that the gap between the spectra of A_{μ} and Â_{μ} in the spectrum of the block operator matrix

( \begin{matrix}A_{μ} & 0 \\ 0 & −A_{μ} \end{matrix} )

is retained as a gap in the spectrum of D_{μ}, independently of the “size” of C. This implies an estimate of the smallest positive eigenvalue of each operator D_{μ} and, by taking the minimum over all μ, also an estimate of the smallest positive eigenvalue of the operator D_{M}.

The main difference to [14] is that there the operators D_{μ} lead to systems of ordinary differential equations on the basis interval I which could be reduced to certain Riccati differential equations, whereas in the more general case of an m– dimensional basis B considered here one is lead to systems of partial differential equations to which we apply the operator theoretic result mentioned above.

The paper is organized as follows. In Section 2 we introduce warped products with boundary and we derive the decomposition of the space of spinors on warped products. In Section 3 we express the Dirac operator D_{M_{0}} on a warped product M_{0} = B_{0} ×_{f} S^{k} formally by means of Dirac operators on the basis B and the fiberwise Dirac operator D′. In Section 4 we establish the boundary conditions for the operators in (1.2) which follow from the condition that a spinor over the warped product B_{0}×_{f} S^{k} has to have a continuous extension to M = B×_{f} S^{k}, more exactly, D_{M} is the closure of the restriction of D_{M_{0}} to all spinors over B_{0} ×_{f} S^{k} with this property. In Section 5 we present the operator theoretic result about the spectral gap of certain block operator matrices. In Section 6 we establish the direct sum decomposition of the above mentioned restriction of D_{M_{0}} in block operator matrices satisfying the assumptions of Section 5. Applying the abstract theorem of Section 5 to the block operator matrices in this decomposition, we obtain our main result about the eigenvalues of the Dirac operator D_{M} in Section 7. Finally, we prove that our result is sharp by showing that for the limiting case of an example it coincides with a well-known upper bound for the first positive eigenvalue of the Dirac operator on an immersed Riemannian manifold.

2. The decomposition of the space of spinors

The manifolds considered in this paper are warped products over a basis manifold B with boundary which has a certain metric.

Definition 2.1. A Riemannian manifold B with boundary ∂B is said to have a warped collar if there exists a neighbourhood U of ∂B of the form

U = ∂B_{g}× [0, ε) := (∂B × [0, ε), g^{2}m_{0} + dt^{2})

where ε > 0, m_{0} is a metric on ∂B and g : [0, ε) → ℝ^{+} is the restriction of an even C^{∞}–function on (−ε, ε) with g(0) = 1.

Lemma 2.2. Let B be an m–dimensional Riemannian manifold with boundary and warped collar U = ∂B_{g}× [0, ε). Let f : B → ℝ_{0}^{+} be a C^{∞}–function with

f|_{B_{0}} > 0, f|_{∂B} ≡ 0

and such that on U the function f(x, ·) = f does not depend on x ∈ ∂B and is the restriction of an odd C^{∞}–function on (−ε, ε) with f′(0) = 1. Then the closure of the warped product M_{0} := B_{0} ×_{f} S^{k} is an (m + k)–dimensional manifold denoted by B ×_{f} S^{k}.

Proof. A neighbourhood of ∂B ⊂ B ×_{f} S^{k} is given by ∂B g̃× (D_{ε}^{k+1}, m_{f}) where g̃ : D_{ε}^{k+1} → ℝ^{+} is given by g̃(x) := g(|x|) and the metric m_{f} on the ε–disc D_{ε}^{k+1} is determined by (D_{ε}^{k+1}, m_{f}) = [0, ε) ×_{f} S^{k}.

Definition 2.3. Let B and f be given as in Lemma 2.2. Then the manifold B×_{f} S^{k} is called warped product with boundary (with basis B and warpe function f). If B ×_{f} S^{k} is a spin manifold, we call B ×_{f} S^{k} a spin warped product with boundary.

We consider M_{0} as a fiber bundle over B_{0} with the canonical connection and we use the notations fiber, horizontal and vertical vectors correspondingly.

For k > 1, we have H_{1}(B ×_{f} S^{k}) = H_{1}(B ×_{f} S^{k} \ ∂B) and hence the spin structures of B ×_{f} S^{k} are in one-to-one correspondence with the spin structures of B_{0} ×_{f} S^{k} and therefore with the spin structures on B_{0}.

The spinor calculus for warped products with 1–dimensional basis has been discussed in [6, p. 18] and [5], the formulas for the Clifford multiplication on products have been established in [8] and more explicitly by Kramer [13]. In [11] the spinor calculus on products has been worked out in some special dimensions. In the sequel we summarize these and present the spinor calculus on warped products M_{0} = B_{0} ×_{f} S^{k} over an m–dimensional basis B. We use the following notations:

Notation. For a Riemannian spin manifold X we denote by P_{X} the frame bundle over X, by Q_{X} a spin structure of P_{X} and by Σ_{X} the corresponding spinor bundle.

Let π_{1} : M_{0} → B_{0} and π_{2} : M_{0} → S^{k} denote the canonical projections. Then the frame bundle P_{M_{0}} has a reduction π_{1}^{∗}P_{B_{0}} ×_{M_{0}} P′ to the structure group SO(m) × SO(k) where P′ =^{∼} π_{2}^{∗}P_{S^{k}} is the bundle of vertical orthogonal frames. Therefore a spin structure Q_{B_{0}} on B_{0} induces a spin structure Q_{M_{0}} on M_{0} by

Q_{M_{0}} = (π_{1}^{∗}Q_{B_{0}} ×_{M_{0}} Q′) ×_{Spin(m)×Spin(k)} Spin(m + k)

where is Q′ =^{∼} π_{2}^{∗}Q_{S^{k}} is a Spin(k)–version of P′. We denote by Σ′ → M_{0} the fiberwise spinor bundle given by Q′ ×_{Spin(k)} Δ_{k}. Then Σ′ =^{∼} π_{2}^{∗}Σ_{S^{k}} and

Σ_{M}|_{M_{0}} = π_{1}^{∗}Σ_{B_{0}} ⊗ Σ′ =^{∼} π_{1}^{∗}Σ_{B_{0}} ⊗ π_{2}^{∗}Σ_{S^{k}} for m or k even,
Σ_{M}|_{M_{0}} = (π_{1}^{∗}Σ_{B_{0}} ⊗ Σ′) ⊕ (π_{1}^{∗}Σ̂_{B_{0}} ⊗ Σ′) =^{∼} (π_{1}^{∗}Σ_{B_{0}} ⊗ π_{2}^{∗}Σ_{S^{k}}) ⊕ (π_{1}^{∗}Σ̂_{B_{0}} ⊗ π_{2}^{∗}Σ_{S^{k}})

for m and k odd.

Here for an n–dimensional manifold X with n odd, we denote by Σ_{X} and Σ̂_{X} the two spinor bundles given by the two representations of the n–dimensional Clifford algebra Cl_{n}.

The Clifford multiplication with vectors X ∈ T B^{m}, Y ∈ T S^{k} on spinors ψ_{1} ⊗ ψ_{2} ∈ Γ_{M_{0}} (π_{1}^{∗}Σ_{B_{0}} ⊗ π_{2}^{∗}Σ_{S^{k}}) and φ_{1} ⊗ φ_{2} ∈ (π_{1}^{∗}Σ̂_{B_{0}} ⊗ π_{2}^{∗}Σ_{S^{k}}) is given by

\begin{aligned}X · (ψ_{1} ⊗ ψ_{2}) & = & X · ψ_{1} ⊗ ψ_{2} \\ Y · (ψ_{1} ⊗ ψ_{2}) & = & ω_{B_{0}} · ψ_{1} ⊗ Y · ψ_{2} \end{aligned}  for m even,

\begin{aligned}X · (ψ_{1} ⊗ ψ_{2}) & = & X · ψ_{1} ⊗ ω_{S^{k}} · ψ_{2} \\ Y · (ψ_{1} ⊗ ψ_{2}) & = & ψ_{1} ⊗ Y · ψ_{2} \end{aligned}  for m odd, k even,

\begin{aligned}X · (ψ_{1}⊗ ψ_{2} + φ_{1}⊗ φ_{2}) & = & i (η^{−1}(X · φ_{1}) ⊗ φ_{2} + η(X · ψ_{1}) ⊗ ψ_{2}) \\ Y · (ψ_{1}⊗ ψ_{2} + φ_{1}⊗ φ_{2}) & = & η^{−1}(φ_{1}) ⊗ Y · φ_{2} + η(ψ_{1}) ⊗ Y · ψ_{2} \end{aligned} for m, k odd.

Here, for an even dimensional manifold A with local orthonormal basis e_{1}, …, e_{a}, ω_{A} := i^{a/2}e_{1} ⋯ e_{a} denotes the complex volume element, and for an odd dimensional manifold A we denote by η : Σ_{A} → Σ̂_{A} the canonical isomorphism with η(v · ψ) = −v · η(ψ).

The Dirac operator D_{S^{k}} on the sphere S^{k} induces an operator D′ on Σ′ = π_{2}^{∗}Σ_{S^{k}} by

(D′ψ)(b, ·) = D_{f(b)S^{k}} (ψ|_{{b}×f(b)S^{k}}) ,

which we call Fiber Dirac operator. For each eigenvalue μ ∈ Spec_{p}(D_{S^{k}}) with multiplicity r(μ) we define an r(μ)-dimensional vector bundle E_{μ} → B_{0} by

(E_{μ})_{b} := E (\frac{μ}{f(b)}, D′|_{\{b\}×f(b)S^{k}})

where E (\frac{μ}{f(b)}, D′|_{\{b\}×f(b)S^{k}}) denotes the eigenspace of the Fiber Dirac opera-

tor restricted to {b} × f(b)S^{k}. Let (σ_{μ,1}, …, σ_{μ,r(μ)}) be an orthonormal basis of E(μ, D_{S^{k}}). Then

(σ′_{μ,1}, …, σ′_{μ,r(μ)}) with σ′_{μ,i} := \frac{π_{2}^{∗}σ_{μ,i}}{f^{k/2}}  (2.1)

is a trivialisation of E_{μ} by fiberwise orthonormal sections. Therefore we get a decomposition of the space of sections

Γ_{M_{0}}(Σ′) = ⨁_{μ∈Spec_{p}(D_{S^{k}})} Γ_{B_{0}}(E_{μ}) = ⨁_{μ∈Spec_{p}(D_{S^{k}})} Γ_{B_{0}}(B_{0} × ℂ^{r(μ)})

as every (continuous) section ψ ∈ Γ_{M_{0}}(Σ′) has a decomposition ψ = ∑_{μ} ϑ_{μ}σ_{μ} with (continuous) functions ϑ_{μ} on B_{0}.

In the same way the spinor space over the warped product M_{0} decomposes as

Γ_{M_{0}}(Σ_{M}|_{M_{0}}) = ⨁_{μ∈Spec_{p}(D_{S^{k}})}Γ_{B_{0}} (Σ_{B_{0}} ⊗ E_{μ}) for m or k even,

Γ_{M_{0}}(Σ_{M}|_{M_{0}}) = ⨁_{μ∈Spec_{p}(D_{S^{k}})} (Γ_{B_{0}}(Σ_{B_{0}} ⊗ E_{μ}) ⊕ Γ_{B_{0}}(Σ̂_{B_{0}} ⊗ E_{μ})) for m and k odd.

Definition 2.4. Let ψ ∈ Γ_{M}(Σ_{M}) and μ ∈ Spec_{p}(D_{S^{k}}). Then ψ is called a spinor of weight μ if

ψ|_{M_{0}} ∈ Γ_{B_{0}} (Σ_{B_{0}} ⊗ E_{μ}) = Γ_{B_{0}}(Σ_{B_{0}} ⊗ ℂ^{r(μ)}) =: W_{μ} if m even,
ψ|_{M_{0}} ∈ Γ_{B_{0}} (Σ_{B_{0}} ⊗ (E_{μ} ⊕ E_{−μ})) = Γ_{B_{0}}(Σ_{B_{0}} ⊗ ℂ^{2r(μ)}) =: W_{μ} if m odd, k even,
ψ|_{M_{0}} ∈ Γ_{B_{0}}((Σ_{B_{0}}^{d}⊗ E_{μ})⊕(Σ_{B_{0}}^{od}⊗ E_{−μ}))= Γ_{B_{0}}((Σ_{B_{0}}^{d}⊗ ℂ^{r(μ)})⊕(Σ_{B_{0}}^{od}⊗ ℂ^{r(μ)}))=:W_{μ}
if m, k odd,

where Σ_{B_{0}}^{d} and Σ_{B_{0}}^{od} are subbundles of Σ_{B_{0}} ⊕ Σ̂_{B_{0}} given by

Σ_{B_{0}}^{d} := \{( \begin{matrix}φ \\ η(φ) \end{matrix} ) : φ ∈ Σ_{B_{0}}\} , Σ_{B_{0}}^{od} := \{( \begin{matrix}φ \\ −η(φ) \end{matrix} ) : φ ∈ Σ_{B_{0}}\} .

The reason for this definition will become clear in the next section. Note that

Γ_{M_{0}}(Σ_{M_{0}}) = ⨁_{μ∈Spec_{p}(D_{S^{k}})} W_{μ}, if m even, or m, k odd,  (2.2)

Γ_{M_{0}}(Σ_{M_{0}}) =  ⨁_{μ∈Spec_{p}(D_{S^{k}})∩ℝ^{+}} W_{μ}, if m odd, k even.  (2.3)

3. The Dirac equation on the warped product

In this section we calculate the Dirac operator on M_{0} in terms of the Dirac operator on B_{0} and the Fiber Dirac operator. For details concerning Dirac operators on Riemannian manifolds see [9] or [16].

First we recall some facts about the Levi–Civita connection ∇ on M_{0}. We denote the Levi–Civita connection on B_{0} by ∇^{B_{0}} and by ∇^{H} : ΓT′M_{0} → Γ (T^{∗}B ⊗ T′M_{0}) the partial covariant derivative in the bundle of vertical vectors given by ∇_{X}^{H}Y := ∇_{X}Y for X ∈ T B_{0}, Y ∈ ΓT′M_{0}. Furthermore, we denote by ∇′ the partial covariant derivative in T′M_{0} which is given by the Levi–Civita connection on the fibers, namely ∇′_{Y_{1}}Y_{2} = ∇_{Y_{1}}^{f(b)S^{k}}Y_{2} with Y_{1}, Y_{2} ∈ ΓT′_{b}M_{0}. Therefore ∇^{H} and ∇′ define together an isometric linear connection on T′M.

The second fundamental form II of the fibers is given by II(Y_{i}, Y_{j}) = −δ_{ij} \frac{gradf}{f} for orthonormal vertical vector fields Y_{i}, Y_{j}. Therefore the classical Gauss formula is

∇_{Y_{i}}Y_{j} = −δ_{ij} \frac{gradf}{f} + ∇′_{Y_{i}}Y_{j}

for orthonormal Y_{i}, Y_{j} ∈ ΓT′M_{0}. From this we obtain, in the case m or k even, for the covariant derivative of a spinor ψ = π_{1}^{∗}ψ_{B_{0}} ⊗ψ′ with ψ_{B_{0}} ∈ ΓΣ_{B_{0}} and ψ′ ∈ ΓΣ′,

∇_{X}(π_{1}^{∗}ψ_{B_{0}} ⊗ ψ′) = π_{1}^{∗}∇_{X}^{B_{0}}ψ_{B_{0}} ⊗ ψ′ + π_{1}^{∗}ψ_{B_{0}} ⊗ ∇_{X}^{H}ψ′ for X ∈ π_{1}^{∗}T B,
∇_{Y} (π_{1}^{∗}ψ_{B_{0}} ⊗ ψ′) = π_{1}^{∗}ψ_{B_{0}} ⊗ ∇′_{Y} ψ′ + \frac{1}{2} \frac{gradf}{f} · Y · (π_{1}^{∗}ψ_{B_{0}} ⊗ ψ′) for Y ∈ π_{2}^{∗}T Σ_{S^{k}}.

Here the induced covariant derivatives in the spinor bundle are denoted with the same symbol as the corresponding operators in the tangential bundle, i.e., ∇^{B_{0}} : ΓΣ_{B_{0}} → Γ(T^{∗}B_{0} ⊗ Σ_{B_{0}}) is induced by the Levi–Civita connection on B_{0} and ∇^{H} : ΓΣ′ → Γ(π_{1}^{∗}(T^{∗}B_{0}) ⊗ Σ′) is induced by the covariant derivative of the Levi– Civita connection on M_{0} of vertical vectors with respect to horizontal vectors. The formulas for m and k odd are completely analogous. Therefore the Dirac operator D_{M_{0}} is given by

D_{M_{0}}(π_{1}^{∗}ψ_{B} ⊗ ψ′) = π_{1}^{∗}D_{B_{0}}ψ_{B} ⊗ ψ′+ π_{1}^{∗} (ω_{B_{0}} · ψ_{B}) ⊗ D′ψ′+ \frac{k}{2} π_{1}^{∗}(\frac{gradf}{f} · ψ_{B})⊗ ψ′

+ ∑_{i=1}^{m} π_{1}^{∗} (e_{i} · ψ_{B}) ⊗ ∇_{e_{i}}^{H}ψ′  for m even,  (3.1)

D_{M_{0}}(π_{1}^{∗}ψ_{B} ⊗ ψ′) = π_{1}^{∗}D_{B_{0}}ψ_{B} ⊗ ω_{fS^{k}}· ψ′+π_{1}^{∗}ψ_{B} ⊗ D′ψ′+ \frac{k}{2} π_{1}^{∗}(\frac{gradf}{f} · ψ_{B})⊗ ω_{fS^{k}}· ψ′

+ ∑_{i=1}^{m} π_{1}^{∗} (e_{i} · ψ_{B}) ⊗ ω_{fS^{k}}· ∇_{e_{i}}^{H}ψ′ for m odd, k even,  (3.2)

D_{M_{0}}(\begin{matrix}π_{1}^{∗}ψ_{B} ⊗ ψ′ \\ π_{1}^{∗}φ_{B} ⊗ φ′ \end{matrix})= i ( \begin{matrix}η^{−1}(D_{B_{0}}φ_{B}) ⊗ φ′ \\ η(D_{B_{0}}ψ_{B}) ⊗ ψ′ \end{matrix} ) + ( \begin{matrix}η^{−1}(φ_{B}) ⊗ D′φ′ \\ η(ψ_{B}) ⊗ D′ψ′ \end{matrix} )

+ \frac{ik}{2} (\begin{matrix}η^{−1}(\frac{gradf}{f} · φ_{B}) ⊗ φ′ \\ η (\frac{gradf}{f} · ψ_{B}) ⊗ ψ′ \end{matrix})+ i ∑_{i=1}^{m}(\begin{matrix}η^{−1}(e_{i} · φ_{B}) ⊗ ∇_{e_{i}}^{H}φ′ \\ η (e_{i} · ψ_{B}) ⊗ ∇_{e_{i}}^{H}ψ′ \end{matrix})

for m and k odd,  (3.3)

where D_{B_{0}} is the Dirac operator on B_{0} and (e_{1}, …, e_{m}) is a local orthonormal basis of T B.

Theorem 3.1. The Dirac operator on M_{0} respects the decomposition (2.2), (2.3) of the space of spinors in spinors of weight μ,

D_{M_{0}} = ⨁_{μ} D_{μ}

with D_{μ} := D_{M_{0}}|_{W_{μ}} given as follows:

1) For m even

D_{μ} = ( \begin{matrix}μ/f & D_{B}^{+} \\ D_{B}^{−} & −μ/f \end{matrix} )

with respect to the decomposition W_{μ} = Γ_{B_{0}}(Σ_{B_{0}}^{+} ⊗ E_{μ}) ⊕ Γ_{B_{0}}(Σ_{B_{0}}^{−} ⊗ E_{μ}).

2) For m odd and k even

D_{μ} = ( \begin{matrix}μ/f & D_{B} \\ D_{B} & −μ/f \end{matrix} )

on W_{μ} = Γ_{B_{0}} (Σ_{B_{0}} ⊗ E_{μ}) ⊕ Γ_{B_{0}} (Σ_{B_{0}} ⊗ E_{−μ}).

3) For m odd and k even

D_{μ} = ( \begin{matrix}μ/f & iD_{B} \\ −iD_{B} & −μ/f \end{matrix} )

on W_{μ} = Γ_{B_{0}} (Σ_{B_{0}}^{d} ⊗ E_{μ}) ⊕ Γ_{B_{0}} (Σ_{B_{0}}^{od} ⊗ E_{−μ}), where here D_{B} denotes the sum of the Dirac operators on Σ_{B_{0}} and Σ̂_{B_{0}}.

Proof. We use the relation

∇_{e_{i}}^{H} \frac{π_{1}^{∗}ψ_{S^{k}}}{f^{k/2}} = −\frac{k}{2} \frac{df(e_{i})}{f} \frac{π_{1}^{∗}ψ_{S^{k}}}{f^{k/2}}

for ψ_{S^{k}} ∈ ΓΣ_{S^{k}}. Then the representation in 1) follows from (3.1) and from ω_{B}|_{Σ_{B}^{±}} = ±id|_{Σ_{B}^{±}}. The formula in 2) is a consequence of (3.2) and the fact that ω_{S^{k}} is an isomorphism from E(μ, D_{S^{k}}) to E(−μ, D_{S^{k}}). In the case 3) when m and k are odd we use the fact that the isomorphism η anti-commutes with the Dirac operators on B_{0}. Thus (3.3) gives

D_{M_{0}} (( \begin{matrix}ψ_{B} \\ ±η(ψ_{B}) \end{matrix} ) ⊗ σ′_{μ,i}) = i ( \begin{matrix}∓D_{B_{0}}ψ_{B} \\ η(D_{B_{0}}ψ_{B}) \end{matrix} ) ⊗ σ′_{μ,i} + ( \begin{matrix}±ψ_{B} \\ η(ψ_{B}) \end{matrix} ) ⊗ \frac{μ}{f} σ′_{μ,i}

where σ′_{μ,i} is defined as in (2.1).

4. The boundary conditions

In the last two section we have considered the space of spinors and the Dirac operator on M_{0} = M \∂B = B_{0} ×_{f} S^{k}. Obviously, not every spinor on M_{0} defines a continuous spinor on M. In the following theorem we establish a necessary condition for a spinor to have a continuous extension to M.

Theorem 4.1. Let ψ be a spinor of weight μ on M_{0},

ψ ∈ Γ_{B_{0}} (Σ_{B^{m}} ⊗ ℂ^{r(μ)})  for m even,
ψ ∈ Γ_{B_{0}} (Σ_{B^{m}} ⊗ ℂ^{2r(μ)})  for m odd, k even,
ψ ∈ Γ_{B_{0}} ((Σ_{B^{m}}^{d} ⊗ ℂ^{r(μ)}) ⊕ (Σ_{B^{m}}^{od} ⊗ ℂ^{r(μ)}))  for m and k odd.

If ψ has a continuous extension to a spinor on M, then for each of the components ψ_{i} of ψ

lim_{t→0} \frac{ψ_{i}(x, t)}{f^{k/2}(t)} exists for (x, t) ∈ ∂B_{g}× (0, ε).  (4.1)

Proof. Let U be a metric collar of ∂B, namely U = ∂B_{g}× [0, ε). Then we have U ×_{f} S^{k} =^{∼} ∂B × D^{k+1}. The restriction of a spinor ψ ∈ Γ_{M_{0}}Σ_{M_{0}} to (U ×_{f} S^{k}) \ ∂B is therefore a spinor over ∂B × (D^{k+1} \ {0}). More exactly, in the case m and k even the image of

ψ = π_{1}^{∗} (θ_{1}ψ_{∂B} + θ_{2}ψ̂_{∂B}) ⊗ \frac{π_{2}^{∗}σ_{i}}{f^{k/2}} ∈ π_{1}^{∗}Σ_{B_{0}} ⊗ π_{2}^{∗}Σ_{S^{k}}

with θ_{i} ∈ C^{0}(0, ε), ψ_{∂B} ∈ ΓΣ_{∂B} and ψ̂_{∂B} ∈ ΓΣ̂_{∂B} under the isomorphism

Σ_{(U×_{f} S^{k})\∂B} =^{∼} (π_{3}^{∗}Σ_{∂B} ⊗ π_{4}^{∗}Σ_{D^{k+1}\{0}}) ⊕ (π_{3}^{∗}Σ̂_{∂B} ⊗ π_{4}^{∗}Σ_{D^{k+1}\{0}})

is given by

(π_{3}^{∗}ψ_{∂B} ⊗ π_{4}^{∗}θ_{1} \frac{σ_{i}}{f^{k/2}} ) ⊕ (π_{3}^{∗}ψ̂_{∂B} ⊗ π_{4}^{∗}θ_{2} \frac{σ_{i}}{f^{k/2}} )

and can be extended continuously to U ×_{f} S^{k} if and only if

θ_{j} \frac{σ_{i}}{f^{k/2}} ∈ ΓΣ_{D^{k+1}\\{0\}} = C^{0} (D^{k+1} \ \{0\}, ℂ^{2^{k/2}}) , j = 1, 2,

can be extended continuously to ΓΣ_{D^{k+1}}. Here π_{3} and π_{4} denote the projections on ∂B and D^{k+1}, respectively. The inclusion S^{k} ↪ ℝ^{k+1} induces an inclusion Σ_{S^{k}} ↪ Σ_{ℝ^{k+1}}, and the image of σ_{i} under this inclusion is a nonzero map on S^{k}. Therefore a necessary condition for θ_{j} \frac{σ_{i}}{f^{k/2}} to be continuously extendable to D^{k+1} is that lim_{t→0} θ_{j} \frac{σ_{i}}{f^{k/2}} exists. In all other cases the assertion is proved analogously.

5. An operator theoretic theorem

In this section we present the operator theoretic theorem established in [15] by means of which we are going to prove eigenvalue estimates for the Dirac operator D_{M}. For the basic operator theoretic notions used in the following we refer the reader e.g. to [12].

We consider a Hilbert space H̃ = H ⊕ Ĥ which is the orthogonal sum of two Hilbert spaces H and Ĥ. We assume that Ã_{0} is an (unbounded) operator in H̃ which admits a block operator representation

Ã_{0} = ( \begin{matrix}A & C \\ C^{∗} & D \end{matrix} )

with densely defined closable operators A, C and D acting between the respective spaces. The operator Ã_{0} is defined on its natural domain (D(A) ∩ D(C^{∗})) ⊕ (D(C) ∩ D(D)), and we suppose that this domain is dense in H̃ and that Ã_{0} is also closable. The closure of Ã_{0} is denoted by Ã.

We recall that a point z ∈ ℂ is called a point of regular type of Ã if there exists a constant c_{z} > 0 such that

∥(Ã − z)x̃∥ ≥ c_{z}∥x̃∥, x̃ ∈ D(Ã).

The set of points of regular type of Ã is denoted by r(Ã). Obviously, if z is a point of regular type of Ã, then Ã − z is injective and hence z ∉ Spec_{p}(Ã).

Theorem 5.1. Suppose the block operator matrix Ã_{0} is densely defined and closable with closure Ã and that there exist positive numbers α, δ such that

\begin{aligned}ℜ (Ax, x) & ≥ & α ∥x∥^{2}, & x ∈ D(A) ∩ D(B^{∗}), \\ ℜ (Dx̂, x̂) & ≤ & −δ ∥x̂∥^{2}, & x̂ ∈ D(B) ∩ D(D). \end{aligned}  (5.1)

Then the strip {z ∈ ℂ : −δ < ℜ(z) < α} belongs to the set r(Ã) of points of regular type of Ã. In particular,

Spec_{p}(Ã) ∩ {z ∈ ℂ : −δ < ℜ(z) < α} = ∅.

Proof. The theorem has been proved in [15] under the assumption that the operators A, C and D are closed (see Theorem 2.1 therein). The proof carries over without changes to the case that A, C and D are only closable, but we repeat it for the convenience of the reader.

Let z ∈ ℂ be such that −δ < ℜ(z) < α. In order to show that z ∈ r(Ã), it is sufficient to show that there exists a constant c_{z} > 0 such that

∥(Ã_{0} − z)x̃∥ ≥ c_{z}∥x̃∥, x̃ ∈ D(Ã_{0}).  (5.2)

For if ỹ ∈ D(Ã) is arbitrary, then, since Ã is the closure of Ã_{0}, there exists a sequence (x̃_{n})_{1}^{∞} ⊂ D(Ã_{0}) such that x̃_{n} → ỹ, Ã_{0}x̃_{n} → Ãỹ, n → ∞ (see [12], Chapter III, § 5.3). By (5.2), we know that ∥(Ã_{0} − z)x̃_{n}∥ ≥ c_{z}∥x̃_{n}∥, n ∈ ℕ. Taking the limit on both sides, we find that ∥(Ã − z)ỹ∥ ≥ c_{z}∥ỹ∥, that is, z ∈ r(Ã).

Assume now that there exists no constant c_{z} > 0 with (5.2). Then there exists a sequence (x̃_{n})_{1}^{∞}, ∥x̃_{n}∥ = 1, such that (Ã_{0}−z)x̃_{n} → 0, n → ∞. With x̃_{n} = (x_{n}, x̂_{n})^{t} we obtain

(A − z)x_{n} + Bx̂_{n} → 0, B^{∗}x_{n} + (D − z)x̂_{n} → 0, n → ∞,

which implies

((A − z)x_{n}, x_{n}) + (Bx̂_{n}, x_{n}) → 0, (B^{∗}x_{n}, x̂_{n}) + ((D − z)x̂_{n}, x̂_{n}) → 0, n → ∞.

If we form the difference of these relations it follows that

((A − z)x_{n}, x_{n}) − ((D − z)x̂_{n}, x̂_{n}) + (Bx̂_{n}, x_{n}) − \overline{(Bx̂_{n}, x_{n})} → 0, n → ∞,

whence

ℜ (Ax_{n}, x_{n}) − ℜ(z)∥x_{n}∥^{2} − ℜ (Dx̂_{n}, x̂_{n}) + ℜ(z)∥x̂_{n}∥^{2} → 0, n → ∞.

By the assumptions (5.1), the left hand side is bounded from below by

(α − ℜ(z)) ∥x_{n}∥^{2} + (δ + ℜ(z)) ∥x̂_{n}∥^{2},

which is strictly positive since −δ < ℜ(z) < α and ∥x̃_{n}∥^{2} = ∥x_{n}∥^{2} + ∥x̂_{n}∥^{2} = 1, a contradiction.

6. The Dirac operator on M

The Dirac operator D_{M} on a spin warped product M with boundary in the Hilbert space L_{2}(M, Σ_{M}) with domain D(D_{M}) = W_{2}^{1}(M, Σ_{M}) (the first order Sobolev space) is self-adjoint because M is a closed Riemannian manifold (see [18]).

Denote by Γ_{ex}^{∞}(M_{0}, Σ_{M_{0}}) the set of all C^{∞}–spinors on M_{0} which possess a continuous extension to M. Then D_{M} is the closure of the restriction of the Dirac operator on M_{0} to Γ_{ex}^{∞}(M_{0}, Σ_{M_{0}}):

D_{M} = \overline{D_{M_{0}}|_{Γ_{ex}^{∞}(M_{0},Σ_{M_{0}} )}}.

Hence D_{M_{0}}|_{Γ_{ex}^{∞}(M_{0},Σ_{M_{0}} )} is essentially self-adjoint.

Theorem 6.1. For μ ∈ Spec_{p}(D_{S^{k}}) we consider the Hilbert space H̃_{μ} = H_{μ} ⊕ Ĥ_{μ} where

H_{μ} ⊕ Ĥ_{μ} = \{ \begin{matrix}L_{2}(Σ_{B}^{+} ⊗ ℂ^{r(μ)}) ⊕ L_{2}(Σ_{B}^{−} ⊗ ℂ^{r(μ)}) & for m even, \\ L_{2}(Σ_{B} ⊗ ℂ^{r(μ)}) ⊕ L_{2}(Σ_{B} ⊗ ℂ^{r(μ)}) & for m odd, k even, \\ L_{2}(Σ_{B}^{d} ⊗ ℂ^{r(μ)}) ⊕ L_{2}(Σ_{B}^{od} ⊗ ℂ^{r(μ)}) & for m and k odd, \end{matrix}

and for simplicity we write L_{2} and W_{2}^{1} for L_{2}-spaces and first order Sobolev spaces when the underlying space determined by H_{μ} or Ĥ_{μ} is clear. We introduce the operators

A_{μ} : H_{μ} → H_{μ}, D(A_{μ}) = \{Ψ_{1} ∈ H_{μ} : \frac{1}{f} Ψ_{1} ∈ L_{2}\} , A_{μ}Ψ_{1} = \frac{μ}{f}Ψ_{1},

Â_{μ} : Ĥ_{μ} → Ĥ_{μ}, D(Â_{μ}) = \{Ψ_{2} ∈ Ĥ_{μ} : \frac{1}{f} Ψ_{2} ∈ L_{2}\} , A_{μ}Ψ_{2} = \frac{μ}{f}Ψ_{2},

and

C : Ĥ_{μ} → H_{μ}, D(C) = {Ψ_{2} ∈ Ĥ_{μ} : Ψ_{2} ∈ W_{2}^{1}, Ψ_{2}|_{∂B} = 0},

CΨ_{2} = \{ \begin{matrix}D_{B}^{+}Ψ_{2} & for m even, \\ D_{B}Ψ_{2} & for m odd, k even, \\ iD_{B}Ψ_{2} & for m and k odd. \end{matrix}

Let k > 1. Then the operator D_{M_{0}}|_{Γ_{ex}^{∞}(M_{0},Σ_{M_{0}} )} has a representation

D_{M_{0}}|_{Γ_{ex}^{∞}(M_{0},Σ_{M_{0}} )} = (⨁_{μ} Ã_{μ})|_{Γ_{ex}^{∞}(M_{0},Σ_{M_{0}} )} .  (6.1)

with essentially self-adjoint operators Ã_{μ} in H̃_{μ} which are given by a block operator representation

Ã_{μ} = ( \begin{matrix}A_{μ} & C \\ C^{∗} & −Â_{μ} \end{matrix} )

with domain D(Ã_{μ}) given by (D(A_{μ}) ∩ D(C^{∗})) ⊕ (D(Â_{μ}) ∩ D(C)), i.e.,

\{Ψ_{1} ∈ H_{μ} : Ψ_{1} ∈ W_{2}^{1}, \frac{1}{f} Ψ_{1} ∈ L_{2}\}⊕ \{Ψ_{2} ∈ Ĥ_{μ} : Ψ_{2} ∈ W_{2}^{1}, Ψ_{2}|_{∂B} = 0, \frac{1}{f} Ψ_{2} ∈ L_{2}\} .

Proof. By Theorem 4.1, the components of every spinor ψ ∈ Γ_{ex}^{∞}(M_{0}, Σ_{M_{0}}) satisfy the boundary condition (4.1). This boundary condition first implies that these components vanish at ∂B, i.e. the two components of ψ = (Ψ_{1}, Ψ_{2})^{t} fulfill Dirichlet boundary conditions on the basis manifold B. Secondly, it shows that, if k > 1,

∫_{0}^{ε} |\frac{1}{f(t)}ψ_{i}(x, t)|^{2}g(t) dt = ∫_{0}^{ε} |\frac{ψ_{i}(x, t)}{f^{k/2}(t)}|^{2}f^{k−2}(t) g(t) dt ≤ c ∫_{0}^{ε} f^{k−2}(t) dt < ∞

with a constant c > 0 uniformly in x, i.e. \frac{1}{f} Ψ^{i} ∈ L_{2}, i = 1, 2, due to the assumptions on the warpe functions g and f (see Definition 2.1 and Lemma 2.2).

7. The eigenvalue estimate

Definition 7.1. An eigenvalue of the Dirac operator on a warped product with boundary is called eigenvalue of weight μ with μ ∈ Spec_{p}(D_{S^{k}}) ∩ ℝ^{+} if μ is the smallest positive number such that the corresponding eigenspace contains a spinor of weight μ or −μ.

Theorem 7.2. Let M = B ×_{f} S^{k} be a warped product with boundary with k > 1. Then the spectrum of the Dirac operator on M is symmetric with respect to 0, and if λ is an eigenvalue of weight μ we have the estimate

|λ| ≥ \frac{μ}{f_{max}}

where f_{max} := max{f(x) : x ∈ B}. In particular, the first positive eigenvalue λ_{0} satisfies

λ_{0} ≥ \frac{k}{2f_{max}} .

Proof. The first assertion is immediate if m is even and if m and k are odd. If m is odd and k is even, then if (ψ_{B,1}, ψ_{B,2})^{t} ∈ W_{μ} = Γ ((Σ_{B} ⊗ ℂ^{r(μ)}) ⊕ (Σ_{B} ⊗ ℂ^{r(μ)})) is an eigenspinor to the eigenvalue λ, then (−ψ_{B,2}, ψ_{B,1})^{t} ∈ W_{μ} is an eigenspinor to the eigenvalue −λ.

For the second assertion we note that if λ is an eigenvalue of D_{M}, i.e. of the closure of the operator D_{M_{0}}|_{Γ_{ex}^{∞}(M_{0},Σ_{M_{0}} )}, then it is also an eigenvalue of the operator

A := ⨁_{μ} Ã_{μ}

in the representation (6.1). In particular, if λ is an eigenvalue of weight μ > 0 of D_{M}, then it is an eigenvalue of the operator Ã_{μ}. The operator Ã_{μ} satisfies the assumptions of Theorem 5.1 with

α = δ = \frac{μ}{f_{max}}

since, for Ψ^{1} ∈ D(A_{μ}),

(A_{μ}Ψ^{1}, Ψ^{1})_{L_{2}} = μ ∫_{B} |\frac{Ψ^{1}}{f} |^{2} dB ≥ \frac{μ}{f_{max}} ∥Ψ^{1}∥_{L_{2}}^{2},

and analogously for Â_{μ}. Hence the second assertion ensues from Theorem 5.1. Finally, the last assertion follows from the fact that every eigenvalue of D_{M} is an eigenvalue of one of the operators Ã_{μ} and

min_{μ}\{|λ| : λ ∈ Spec_{p}(Ã_{μ})\} ≥ min \{\frac{|μ|}{f_{max}} : μ ∈ Spec_{p}(D_{S^{k}})\} ≥ \frac{k}{2f_{max}} .

This completes the proof.

In the following we show by means of an example that the lower bound for the first positive eigenvalue in the preceding theorem is sharp.

To this end, let L > 0, and let D_{L}^{m} be the m-dimensional disc with radius L and U_{L}^{m} := [L, L + \frac{π}{2} ] ×_{sin(t−L)+L} S^{m−1}. Gluing these two manifolds, we set

B_{L}^{m} := D_{L}^{m} ⨿_{S_{L}^{m−1}} U_{L}^{m}.

Further, we define the warpe function

f : B_{L}^{m} → ℝ

by

f|_{D_{L}^{m}} ≡ 1, f(t, x) := cos(t − L), (t, x) ∈ U_{L}^{m}.

On M_{L} := B_{L}^{m} ×_{f} S^{k} we consider the isometrical immersion

M_{L} → ℝ^{m+k+1},
(b, y) ↦ (b, ι_{k}(y)), b ∈ D_{L}^{m}, y ∈ S^{k},
(t, x, y) ↦ ((sin(t − L) + L)ι_{m−1}(x), cos(t − L)ι_{k}(y)), (t, x) ∈ U_{L}^{m}, y ∈ S^{k},

where ι_{r} : S^{r} → ℝ^{r+1} denotes the canonical immersion.

For an is an n–dimensional closed oriented hypersurface M isometrically immersed in ℝ^{n+1} with mean curvature H in ℝ^{n+1}, the following estimate for the smallest eigenvalue in modulus of the Dirac operator on M holds (see [4], [7]):

λ^{2} ≤ \frac{n^{2} ∫_{M} H^{2}dM}{4 vol M} .  (7.1)

In the sequel, we calculate the upper bound (7.1) for the smallest positive eigenvalue of the Dirac operator D_{M_{L}} on M_{L} in the limit L → ∞.

The mean curvature H on U_{L}^{m} ×_{f} S^{k} depends only on t ∈ [L, L + π/2], and for the function h given by H(t, x) = h(t − L), t ∈ [L, L + π/2], x ∈ S^{m−1} × S^{k}, we have

h(t) = \frac{1}{m + k} (k + 1 + (m − 1)\frac{sin t}{sin t + L}) .

Therefore the numerator of (7.1) is given by

(m + k)^{2} ∫_{M_{L}} H^{2}dM_{L} = (m + k)^{2} (\frac{k^{2}}{(m + k)^{2}} L^{m}vol D^{m} vol S^{k}

+ ∫_{0}^{π/2}(cos t)^{k}(sin t + L)^{m−1}h^{2}(t) dt vol S^{m−1} vol S^{k})

= k^{2}L^{m}vol D^{m}vol S^{k} + O(L^{m−1}),

and the denominator of (7.1) is given by

4 vol M_{L} = 4L^{m}vol D^{m}vol S^{k} + O(L^{m−1}).

The following remark, which we obtain from (7.1) together with the above calculations and from the fact that f_{max} = 1 here, shows that the lower bounds of Theorem 7.2 are sharp in the limiting case L → ∞.

Remark 7.3. The first positive eigenvalue λ_{0}(L) of the Dirac operator on M_{L} satisfies the estimate

λ_{0}(L)^{2} ≤ \frac{k^{2}}{4} + O(L^{−1}).

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NWF I – Mathematik, University of Regensburg, D-93040 Regensburg, Germany

E-mail address: margarita.kraus@mathematik.uni-regensburg.de

Department of Mathematics and Computer Science, University of Leicester, University Road, Leicester LE1 7RH, United Kingdom

E-mail address: c.tretter@mcs.le.ac.uk