On the Spectral Flow of Families of Dirac Operators with Constant Symbol

Ulrich Bunke^{∗}

April 7, 1993

Abstract

We consider families of generalized Dirac operators D_{t} with constant principal symbol and constant essential spectrum such that the endpoints are gauge equivalent, i.e. D_{1} = W^{∗}D_{0}W. The spectral flow in any gap in the essential spectrum we express as the Fredholm index of 1+(W −1)P where P is the spectral projection on the interval [d, ∞) with respect to D_{0} and d is in the gap. We reduce the computation of this index to the Atiyah-Singer index theorem for elliptic pseudodifferential operators. We find an invariant of the Riemannian geometry for odd dimensional spin manifolds estimating the length of gaps in the spectrum of the Dirac operator.

1 Introduction

If D_{t}, t ∈ [0, 1], is a family of selfadjoint operators, d ∈ R \ (σ(D_{0}) ∪ σ(D_{1})) and d ∉ σ_{ess}(D_{t}), ∀t ∈ [0, 1], then the spectral flow sf(D_{t}) at d is defined as the number of families of eigenvalues λ(t) of D_{t} crossing d from above minus the number of families of eigenvalues crossing d from below.

In the famous paper [1] the spectral flow of a closed loop of Dirac operators D_{t} on a compact manifold M at any d has been expressed as the index of a certain Dirac operator over M × S^{1}. The η-invariant and K-theoretic arguments have been employed there. These techniques are well suited to compact manifolds.

In the present paper we propose a new more functional analytic approach to the spectral flow. It has the advantage that it allows to compute the spectral flow in the presence of essential spectrum, i.e. for families of Dirac operators on noncompact manifolds. The price we have to pay is the restriction to rather special families which we are going to describe now.

Let D be a generalized Dirac operator on a complete Riemannian manifold M associated to a Clifford bundle E → M (See [17] and [21] for definitions. With dom D := C_{c}^{∞}(M, E) the operator D is essentially selfadjoint.) and W ∈ Γ(M, U(E)) be a gauge transformation satisfying

∗Institut für reine Mathematik der Humboldt-Universität zu Berlin, Ziegelstraße 13a, Berlin O-1040, GERMANY, E-mail: bunke@informatik.hu-berlin.de

Assumption 1 1. W − 1 ∈ C_{c}^{∞}(M, End(E))

2. R := W^{∗}DW − D is of zero order.

Let us consider the family D_{t} := D + tR. Then σ_{ess}(D_{t}) = σ_{ess}(D), ∀t ∈ R, and if d ∈ R \ σ(D) then the spectral flow of {D_{t}}_{t=0}^{1} at d is expressed as the Fredholm index of 1 + (W − 1)P where P := E_{D}[d, ∞) and E_{D}(.) denotes the family of spectral projections of D. The operator 1 + (W − 1)P can be localized near the diagonal without changing the index. Then it will be approximated by an elliptic pseudodifferential operator with constant symbol at infinity [6]. The index computation of such operators can be reduced to an index problem on a compact manifold and hence to the Atiyah-Singer index theorem along the lines of [6], §4.1. If D is the twist of the usual Dirac operator on a Riemannian spin manifold M with some Hermitian vector bundle V and W comes from a gauge transformation of V then we end up with the formula

ind(1 + (W − 1)P) = (−1)^{k}CS(W)Â(M)[M]  (1)

where dim M = n = 2k + 1 is odd, CS(W) ∈ H_{c}^{odd}(M, Q) is the Chern-Simons class repesented by the compactly supported form

CS(W) := \frac{i}{2π} T r W^{∗}∇W ∫_{0}^{1} exp (\frac{i(t − t^{2})}{2π}W^{∗}∇W W^{∗}∇W ) dt  (2)
CS(W)_{2r−1} = 2\frac{i^{r}}{(2π)^{r}} \frac{r!}{(2r)!}T r([W^{∗}∇W]^{2r−1}),  (3)

Â(M) is the Â-genus and [M] is the fundamental cycle of M. If the dimension of M is even then

ind(1 + (W − 1)P) = 0.  (4)

Because of (1) the spectral flow

sf{D_{t}} = ind(1 + (W − 1)P)  (5)

does not depend on the gap containing d. A similar formula has been obtained in [12] for the spectral flow of a family of Dirac operators on a compact manifold with boundary.

In the present paper we produce by geometric means many examples where (1) does not vanish. In [9] a triple (D, R, [0, 1]) is called complete if for any λ ∈ R there is a t ∈ [0, 1] such that λ ∈ σ(D + tR). In that reference the authors consider the case where D is the laplace operator and R is a potential. The question of completeness turns out to be very complicated in that case. As we shall see in the present paper for Dirac operators to show completeness is sometimes much easier when one can apply index theory.

Another application is very similar to the ideas of [23]. Let ω_{M} ∈ H_{c}^{n}(M, Q) be the dual of the fundamental cycle. We define the Riemannian invariant for odd-dimensional spin manifolds M

G(M)^{2} := inf sup_{M} tr(W^{∗}D_{V} W − D_{V})^{2}  (6)

where the infimum is taken over all trivial Hermitian vector bundles with the flat connection V and gauge transformations W of V satisfying Assumption 1 and CS(W) = cω_{M} with some c ≠ 0. Here D is the Dirac operator associated to the spinor bundle S(M) and D_{V} is associated to the tensor product S(M) ⊗ V. We find out that G(M) estimates the length of gaps in the spectrum of D from above.

On one hand if G(M) = 0 then we have σ(D) = R. By geometric arguments we can show that G(M) = 0 if M is simply connected and has nonpositive sectional curvature. Hence in this case there is no gap in the spectrum of the Dirac operator.

On the other hand, if there is a gap of length l in σ(D) then G(M) ≥ l. In particular, if M has nonnegative scalar curvature s then

G(M)^{2} ≥ \frac{n s_{0}}{n − 1}  (7)

where n := dim M is odd and s_{0} = inf_{m∈M} s(m). Here we employ a lower bound of the spectrum of D^{2} obtained by the method of [16]. On compact manifolds there holds G(M) > 0 since the spectrum of D is pure point and has, of course, gaps. It seems interesting to study the invariant G(M) in more detail.

For completeness we will discuss the even-dimensional case simultaneously.

The auther wishes to thank T. Hirschmann for his interest in this work and useful hints. After this paper was finished the author was informed by S. Hurder about the papers [19],[8] which contain closely related ideas.

2 Perturbation theory and relative index of projections

Let R ∈ C_{c}^{∞}(M, End(E)) be a selfadjoint bundle endomorphism of E. We consider the family D_{t} = D + tR, t ∈ R. Then D_{t} is essentially selfadjoint on dom D_{t} = C_{c}^{∞}(M, E). By the decomposition principle [13], [10] we have σ_{ess}(D_{t}) = σ_{ess}(D), ∀t ∈ R.

Let us fix d ∈ R \ σ(D_{t}) and let P (t) := E_{D_{t}}[d, ∞) be the spectral projection of D_{t} on the interval [d, ∞).

Lemma 2.1 For ϕ ∈ dom D̄_{t} there holds

P (t)ϕ = \frac{ϕ}{2} + lim_{S→∞} \frac{1}{2π} ∫_{−S}^{S}(D_{t} − d + iλ)^{−1}ϕdλ  (8)
= \frac{ϕ}{2} + \frac{1}{π} ∫_{0}^{∞}(D_{t} − d)[(D_{t} − d)^{2} + λ^{2}]^{−1}ϕdλ.  (9)

The integrals are strongly convergent.

Proof: The assertion is a consequence of the identity

θ(x) = \frac{1}{2} + lim_{S→∞} \frac{1}{2π} ∫_{−S}^{S}(x + iλ)^{−1}dλ if x ≠ 0  (10)

and of the spectral theorem. □

Lemma 2.2 1. Let d ∈ R \ σ(D_{t}) ∩ res(D_{s}). Then P (t) − P (s) is compact.

2. If d ∈ ∩_{t∈[a,b]}(R \ σ(D_{t})) then the function t ∈ [a, b] → P (t) is operator norm continuous.

Proof: We have

P (t) − P (s) = \frac{1}{2π} ∫_{−∞}^{∞} [(D_{t} − d + iλ)^{−1} − (D_{s} − d + iλ)^{−1}] dλ  (11)

where the integral converges in the operator norm. In fact by the resolvent identity

(D_{t} − d + iλ)^{−1} − (D_{s} − d + iλ)^{−1} = (D_{t} − d + iλ)^{−1}(s − t)R(D_{s} − d + iλ)^{−1}. (12)

Hence the norm of the integrand is bounded by C|s − t|∥R∥(1 + |λ|)^{−2}. Moreover, by Rellich’s theorem the composition

L^{2}(M, E) ⟶^{(D_{s}−d+iλ)^{−1}} H^{1}(M, E) →^{R} L^{2}(M, E)  (13)

is compact. This proves the first assertion. By H^{1}(M, E) we denote the Sobolev space dom D̄ with the norm ∥ψ∥_{H^{1}}^{2} = ∥ψ∥^{2} + ∥D̄ψ∥^{2}. The second follows from the estimate

∥P (t) − P (s)∥ ≤ C|s − t|.  (14)

□

Let d ∈ R \ σ(D) and P := P (0). If M is compact then P is a pseudodifferential operator of zero order. Hence it can be localized near the diagonal modulo compact operators. The next proposition shows that this remains true in some sense also for complete M. Let K_{1} ⊂ K ⊂ M be compact subsets such that K contains a neighbourhood of K_{1} and χ, χ_{1} ∈ C^{∞}(M), supp(χ_{1}) ⊂ K_{1}, supp(χ) ⊂ M \ K.

Lemma 2.3 The composition χ_{1}P χ is compact.

Proof: Using χ_{1}χ = 0, the second representation of P in Lemma 2.1 and

(D − d)[(D − d)^{2} + λ^{2}]^{−1} = ∫_{0}^{∞}(D − d)e^{−t[(D−d)^{2}+λ^{2}]}dt  (15)

we find

χ_{1}P χ = \frac{1}{π} (∫ ∫_{I_{1}} + ∫ ∫_{I_{2}}) χ_{1}(D − d)e^{−t[(D−d)^{2}+λ^{2}]}χdtdλ  (16)

where

I_{1} := {(t, λ) ∈ [1, ∞) × [0, ∞)}  (17)
I_{2} := {(t, λ) ∈ [0, 1] × [0, ∞)}.  (18)

There are constants C_{1}, C_{3} < ∞, c_{2}, c_{4} > 0 such that

∥χ_{1}(D − d)e^{−t[(D−d)^{2}+λ^{2}]}χ∥ ≤ C_{1}e^{−c_{2}t[1+λ^{2}]} ∀(t, λ) ∈ I_{1}  (19)
∥χ_{1}(D − d)e^{−t[(D−d)^{2}+λ^{2}]}χ∥ ≤ C_{3}e^{−tλ}e^{−c_{4}/t} ∀(t, λ) ∈ I_{2}.  (20)

The last estimate is an off-diagonal estimate obtained by the finite propagation speed method of [7]. Thus (16) converges with respect to the operator norm. Since the heat operator is smoothing and χ_{1} restricts to a compact set, the integrand is compact by Rellich’s theorem. This proves the Lemma. □.

Let χ_{2} ∈ C_{c}^{∞}(M) such that suppχ_{2} ⊂ K. Then using a parametrix construction one can find a pseudodifferential operator (in the local sense) A approximating χ_{2}P χ modulo compact operators. For the principal symbol of A we get σ_{A}(x, ξ) = χ_{2}(x)p(x, ξ)χ(x) where p(x, ξ) is the projection onto the positive spectral subspace of the Clifford multiplication with iξ. Note that it does not depend on d.

Lemma 2.2 indicates when the difference P (t) − P (s) of projections is compact. This is exactly the situation where one can define the relative index [2],[22].

Let P, Q be projections on a Hilbert space H such that P − Q is compact. We define the relative index of P and Q by

I(P, Q) := ind_{ImQ→ImP} P Q.  (21)

In order to verify that P Q is Fredholm (from Im Q to Im P) we give parametrices. Modulo compact operators

P QQP = P QP ∼ P P P = P  (22)
QP P Q = QP Q ∼ QQQ = Q.  (23)

Lemma 2.4 Let Q(t) be a norm continuous path of projections such that P − Q(t) is compact for t ∈ [0, 1]. Then I(P, Q(0)) = I(P, Q(1)).

Proof: Note that the domain of P Q(t) depends on t. Thus the argument is not standard. By [4], 4.3.3, there is a norm continuous family U(t) of unitaries such that

U^{∗}(t)Q(t)U(t) = Q(0).  (24)

Then

I(P, Q(t)) = ind_{ImU(t)Q(0)→ImP} P U(t)Q(0)U(t)^{∗} = ind_{ImQ(0)→ImP} P U(t)Q(0). (25)

But now t → P U(t)Q(0) is a norm continuous family of Fredholm operators between fixed subspaces of H. Hence I(P, Q(t)) does not depend on t. This proves the Lemma. □

Let R be a finite projection, dim R = n, QR = 0.

Lemma 2.5 I(P, Q + R) = I(P, Q) + n

Proof: If RP = 0 then the assertion is obvious. For the general case we use the double H ⊕ H and extend R, Q, P by 0 on the second factor. Let

R(t) = ( \begin{matrix}cos(t) & sin(t) \\ −sin(t) & cos(t) \end{matrix} ) ( \begin{matrix}R & 0 \\ 0 & 0 \end{matrix} ) ( \begin{matrix}cos(t) & −sin(t) \\ sin(t) & cos(t) \end{matrix} ) .  (26)

Then QR(t) = 0, P R(π/2) = 0 and I(P, Q + R(0)) = I(P, Q + R(π/2)) = I(P, Q) + n by Lemma 2.4. □

Lemma 2.6 Let P, Q be related by an unitary W,i.e P = W^{∗}QW. Then I(P, Q) = ind(1 + (W − 1)Q).

Proof: Note that [Q, W] = W (P − Q) is compact. Hence

I(P, Q) = ind_{ImQ→ImP} P Q  (27)
= ind_{ImQ→ImW^{∗}Q}W^{∗}QW Q
= ind_{ImQ→ImQ}QW Q
= ind(1 − Q + QW Q)
= ind(1 + (W − 1)Q + [Q, W]Q)
= ind(1 + (W − 1)Q)

□

3 Spectral flow

Let us return to families of Dirac operators. Let W be a gauge transformation satisfying Assumption 1, R := W^{∗}DW − D and D_{t} := D + tR. Then D_{t} satisfies the assumptions made in section 3. We are going to introduce the spectral flow of {D_{t}}_{t=0}^{1}.

First note that D_{t} is a holomorphic family of type (A) in the sense of [20], i.e. domD̄_{t} = dom D̄, ∀t ∈ R, and t → D_{t}ψ is holomorphic for all ψ ∈ dom D̄. Let [a, b] ⊂ R \ σ_{ess}(D) be a gap in the essential spectrum and choose d ∈ [a, b] ∩ (R \ σ(D)) and ϵ > 0 such that [d − ϵ, d + ϵ] ⊂ σ(D). Then the eigenvalues λ_{α}(t) ∈ [d − ϵ, d + ϵ], α ∈ J (J being some index set), form holomorphic families with Lipschitz constant bounded by ∥R∥. It is easy to see that J must be finite. Every family is defined on a maximal interval [s_{α}, t_{α}]. The spectral flow of {D_{t}}_{t=0}^{1} is defined by

sf(D_{t}) = ♯{α ∈ J|λ_{α}(s_{α}) > d, λ_{α}(t_{α}) < d} − ♯{α ∈ J|λ_{α}(s_{α}) < d, λ_{α}(t_{α}) > d} (28)

where the families are counted according to their multiplicities. The main result of this section is

Proposition 3.1 sf{D_{t}} = ind(1 + (W − 1)P)

Proof: Choose a finite partition of [0, 1]:

0 = s_{0} < t_{0} = s_{1} < t_{1} … = s_{N} < t_{N} = 1  (29)

and d_{i}, i = 0, …, N, such that d_{i} ∈ res D_{t}, ∀t ∈ [s_{i}, t_{i}], d_{0} = d_{N} = d. Let P (t) = E_{D_{t}}[d_{i}, ∞) for t ∈ [s_{i}, t_{i}). Then by Lemma 2.2 P (t) is continuous on every interval [s_{i}, t_{i}) and

lim_{t→t_{i}} P (t) = P (s_{i+1}) + sign(d_{i+1} − d_{i})R_{i+1}  (30)

where R_{i+1} is the finite projection onto the eigenspaces according to the eigenvalues of D_{s_{i+1}} between d_{i} and d_{i+1}. Hence R_{i+1}P (s_{i+1}) = 0. Moreover P (t) − P (1) is compact since there is no essential spectrum in [a, b]. Hence by Lemma 2.5

I(P (1), P (s_{i})) = I(P (1), P (s_{i+1})) + sign(d_{i+1} − d_{i})dim R_{i+1}.  (31)

Summing up it follows I(P (1), P (0)) = sf{D_{t}}. Finally apply Lemma 2.6. □ Let D_{t} be as in the last section. We have shown that the spectral flow at d is given by

sf{D_{t}} = ind(1 + (W − 1)P)  (32)

where P = E_{D}[d, ∞). We want to compute this index along the lines of [6]. Let K_{1} = supp(W_{1}) and K ⊂ M be compact containing a neigbourhood of K_{1}. Choose χ ∈ C_{c}^{∞}(M) such that χ_{|K} = 1.

Lemma 3.2 Let Ã := (W − 1)P χ + 1. Then Ã − (1 + (W − 1)P) is compact and sf{D_{t}} = ind Ã.

Proof: We have

(1 + (W − 1)P) − Ã = (W − 1)P (1 − χ).  (33)

Since (W − 1)(1 − χ) = 0 we can apply Lemma 2.3. □

Ã has a pseudodifferential approximation A modulo compact operators as described at the end of section 3. A is an elliptic operator with constant symbol at infinity. In [6] the index of such operators was computed.

Theorem 3.3 Let M be a complete Riemannian spin manifold, V be a Hermitian vector bundle with compatible connection and D_{V} be the twisted Dirac operator on S(M) ⊗ V. Moreover let W be a gauge transformation of V with supp(1 − W) compact and D_{t} := (1 − t)D_{V} + tW^{∗}D_{V} W. Choose d ∈ R \ σ(D_{V}) (i.e. d is in a gap of σ_{ess}(D_{V})). Then the spectral flow of D_{t} at d is 0 if dim M is even and

sf{D_{t}} = (−1)^{k}Â(M)CS(W)[M]  (34)

if dim M = 2k + 1 where Â(M) is the Â-genus, [M] is the fundamental cycle of M and CS(W) ∈ H_{c}^{odd}(M, Q) is represented by

CS(W) := \frac{i}{2π} T r W^{∗}∇W ∫_{0}^{1} exp (\frac{i(t − t^{2})}{2π}W^{∗}∇W W^{∗}∇W ) dt  (35)
CS(W)_{2r−1} = 2\frac{i^{r}}{(2π)^{r}} \frac{r!}{(2r)!}T r([W^{∗}∇W]^{2r−1}).  (36)

Proof: Use the formula for the index of A given in [6]. □

4 Application to spectral theory

Let M be a complete Riemannian spin manifold with the spinor bundle S(M) and V := M × C^{N} be the flat Hermitian vector bundle. The twisted Dirac operator D_{V} is isomorphic to the direct sum of N copies of the Dirac operator D associated to S(M). A gauge transformation W ∈ C^{∞}(M, U(V)) with W − 1 ∈ C_{c}^{∞}(M, End(V)) represents an element [W] ∈ K_{1}(C_{0}(M)) ≅ K_{c}^{1}(M). Here K_{1}(C_{0}(M)) is the K_{1}-group of the algebra of continuous functions vanishing at infinity and K_{c}^{1}(M) is the K-theory with compact support. Note that the Chern-Simons class gives an isomorphism of groups

K_{c}^{1}(M)_{Q} →^{CS} H_{c}^{odd}(M, Q).  (37)

This follows from the Kuenneth-formula

K_{c}^{0}(S^{1} × M)_{Q} = pr_{M}^{∗}K_{c}^{0}(M)_{Q} ⊕ K_{c}^{1}(M)_{Q}d

where d ∈ K^{1}(S^{1}) is the generator and

CS([W]) = (pr_{M})_{∗}ch([W]d).

Let dimM = 2r−1 be odd. Then there is a class ω_{M} ∈ H_{c}^{2r−1}(M, Q) dual to [M] and there is a class [W] ∈ K_{c}^{1}(M) with CS([W]) = lω_{M} for some l ≠ 0. Moreover there is a N ∈ N such that [W] is represented by a gauge transformation of the bundle V. We introduce the following Riemannian invariant: Let R(W) := W^{∗}D_{V} W − D_{V} ∈ Γ(M, End(S(M) ⊗ V)). Then we define

G(M)^{2} := inf sup_{M} trR(W)^{2}  (38)

where the infimum is taken over all trivial Hermitian vector bundles V with the flat connection and gauge transformations satisfying Assumption 1 with CS(W) = lω_{M} for some l ≠ 0. We have ∥R(W)∥^{2} = ∥R^{2}(W)∥ = sup_{M} |R(W)^{2}| ≤ sup_{M} trR(W)^{2}.

Theorem 4.1 Let dim M be odd. G(M) is an upper bound of the length of gaps in the spectrum of D.

Proof: Note that the spectrum of D_{V} coincides (as a set) with the spectrum of D. For every ϵ > 0 there is a family {D_{t}}_{t=0}^{1} with D_{0} = D_{V}, having nonvanishing spectral flow and ∥D_{1} − D_{0}∥ ≤ G(M) + ϵ. Assume that there is an interval [a, b] ∈ R with σ(D_{V})∩[a, b] = ∅ and b−a > G(M). According to the sign of the spectral flow there is at least one family of eigenvalues λ(t) crossing a from below or b from above and not crossing back the same boundary. But then for ϵ small enough λ(1) ∈ [a, b] and λ(1) ∈ σ(D_{V}). This is a contradiction. □.

Corollary 4.2 If G(M) = 0 then σ_{ess}(D) = R.

5 THE INVARIANT G(M) 9

Let us now discuss the even-dimensional analog of G(M). On even-dimensional M one employs the relative index theorem [17],[11] and certain splittings of trivial bundles of compact support. Let V := M × C^{N} be a trivial Hermitian vector bundle. A splitting of compact support is a decomposition V := U ⊕ W where W, U are Hermitian vector bundles with compatible connections being trivial and with the flat connection outside of a compact subset of M. Then ch(W), ch(U) ∈ H_{c}^{ev}(M, Q). In this case we set R := D_{V} − D_{U} ⊕ D_{W} ∈ Γ(M, End(S(M) ⊗ V)) and define

G(M)^{2} := inf sup_{M} \frac{1}{2}trR^{2}  (39)

where the infimum is taken over all V and splittings U ⊕ W of compact support such that dim U = dim W and ch(U) = dim U + lω_{M} for some l ≠ 0. Such splittings exist. Again ∥R∥^{2} ≤ \frac{1}{2}trR^{2} (see [3]).

Theorem 4.3 If dim M is even then [−G(M), G(M)] ∩ σ(D) ≠ ∅.

Proof: The argument is similar to that used in [23],[3],[5],[17]. Let ϵ > 0. Choose a splitting V = U ⊕ W such that ch(U) = dim U + lω_{M} for some l ≠ 0, dim U = dim W and

\frac{1}{2} sup_{M} < G(M)^{2} + \frac{ϵ}{2}

for R := D_{V} − D_{U} ⊕ D_{W}. Then ∥R∥^{2} ≤ G(M)^{2} + ϵ/2. Assume now [−G(M) − ϵ, G(M) + ϵ] ∩ σ(D) = ∅. Then 0 ∉ σ(D_{U}), σ(D_{V}). But the relative index theorem of [17] gives

ind(D_{U}) − ind(D_{W}) = Â(M)(ch(U) − ch(V))[M] = 2l ≠ 0.

Thus at least one of D_{W},D_{U} has a non-trivial kernel. This is a contradiction. The claim follows since ϵ was arbitrary. □

In fact the relative index theorem for Dirac operators [17] has been introduced for a similar application. The result in the even-dimensional case is much weaker than that in the odd-dimensional case since only the gap at zero can be estimated.

5 The invariant G(M)

In this section we study the invariant G(M) in more detail. Let M be a complete spin manifold.

Lemma 5.1 If M^{n} has nonnegative scalar curvature s then

G(M)^{2} ≥ \frac{s_{0}n}{(n−1)} if n is odd  (40)
G(M)^{2} ≥ \frac{s_{0}n}{4(n−1)} if n is even  (41)

where s_{0} := inf_{m∈M} s(m).

Proof: Using the Weizenböck formula of [16] it is easy to show that σ(D^{2}) ⊂ [\frac{s_{0}n}{4(n−1)}, ∞). The Lemma follows from Theorem 4.1 and 4.3. □.

Of course for M compact G(M) > 0 in any case.

Now we want to find upper bounds for G(M) in geometric terms. We say that a Riemannian manifold (M^{n}, g) dominates another Riemannian manifold (M̃^{n}, g̃) if there is a map f : M → M̃ of nonvanishing degree such that g ≥ f^{∗}g̃. If M̃ is compact then f has to be constant outside of a compact subset of M. If M̃ is noncompact then we require f to be proper.

Lemma 5.2 If M^{n} dominates M̃^{n} then G(M) ≤ G(M̃).

Proof: f is used to pull back the gauge transformations in the odd-dimensional and the splittings in the even-dimensional case. We consider first the odd-dimensional case. A local computation shows that

R(W) = ∑_{i=1}^{n} ẽ_{i} ⊗ W^{∗}dW (ẽ_{i})  (42)

where {ẽ_{i}}_{i=1}^{n} is a local orthonormal repere of T M̃. Hence

trR^{2}(W) = −2^{[n/2]} ∑_{i=1}^{n} trW^{∗}dW (ẽ_{i})W^{∗}dW (ẽ_{i})  (43)

where we have used trẽ_{i}ẽ_{j} = −2^{[n/2]}δ_{ij}. Let m ∈ M and f(m) =: m̃. Then at m

R(f^{∗}W) = ∑_{i=1}^{n} e_{i} ⊗ W^{∗}dW (df(e_{i}))  (44)

where {e_{i}}_{i=1}^{n} is a local orthonormal frame of T M around m and W, dW is taken at m̃. Hence

trR^{2}(f^{∗}W) = −2^{[n/2]} ∑_{i=1}^{n} trW^{∗}dW (df(e_{i}))W^{∗}dW (df(e_{i})).  (45)

Let df(e_{i}) = A_{ij}ẽ_{j} and B_{ij} := −trW^{∗}dW (ẽ_{i})W^{∗}dW (ẽ_{j}). The condition f^{∗}g̃ ≤ g translates to AA^{∗} ≤ 1 and hence ∥A∥ ≤ 1. Moreover B ≥ 0 since (W^{∗}dW (X))^{∗} = −W^{∗}dW (X) for all X ∈ T M̃. It follows

trR^{2}(f^{∗}W) = 2^{[n/2]}trABA^{∗}  (46)
≤ 2^{[n/2]}trB
= trR^{2}(W).

This proves the Lemma in the odd-dimensional case.

Now we consider the even-dimensional case. If V = U ⊕ W is a splitting and ∇^{U}, ∇^{W} are compatible connections we set Q := ∇^{V} −∇^{U} ⊕∇^{W} ∈ Γ(M̃, T^{∗}M̃ ⊗End(V)). Locally

R := D_{V} − D_{U} ⊕ D_{W} = ∑_{i=1}^{n} ẽ_{i} ⊗ Q(ẽ_{i}).  (47)

To prove the Lemma we use the same argument as above replacing W^{∗}dW by Q. Note that for the pulled back splitting one gets f^{∗}Q and that Q(X)^{∗} = −Q(X) for all X ∈ T M̃. □

Lemma 5.3 G(S^{n}) ≤ 2^{n/2}\sqrt{n} for n odd and G(S^{n}) ≤ 2^{\frac{n−3}{2}} \sqrt{n} for n even.

Proof: If n is even this follows from [3]. If n is odd we construct explicitely a generator of π_{n}(U(2^{(n+1)/2})). Let △^{n+2} be the spinor representation of Spin(n + 2). The Clifford algebra Cliff(R^{n+2}) acts on △^{n+2} in the usual way. Let △_{+}^{n+1} be the −i-eigenspace of the endomorphism N := (0, …, 0, 1) ∈ R^{n+2}. We consider the embedding S^{n} ⊂ S^{n+1} as the equator such that N is the north pole of S^{n+1}. Let e_{1} := (1, 0, …, 0). Then the gauge transformation W is given by η ∈ S^{n} → W (η) := −ie_{1}η ∈ U(△_{+}^{n+1}). One can show by explicit computation that CS(W)[S^{n}] = 1 (Another way to see this is to apply the Dynin-Fedosov formula to the Bott generator W [15],[18]). For this gauge transformation we have tr(W^{∗}D_{V} W − D_{V})^{2} = 2^{n}n. □

For the sphere of sectional curvature K we have G(S^{n}(K)) = \sqrt{K}G(S^{n}(1)). It was shown e.g. in [3] that every complete Riemannian manifold dominates some sphere.

Lemma 5.4 Let K = inf_{p∈M} K(p) with

K(p) = sup_{m∈B(p,r_{inj}(p))} min(K(m) , \frac{π^{2}}{r_{inj}^{2}(p)})  (48)

where r_{inj}(p) is the injectivity radius of M at p ∈ M and K(m) is the maximal sectional curvature at m ∈ M. There is a function α(ϵ) with lim_{ϵ→0} α(ϵ) = 0 such that for every small ϵ > 0 the manifold (M^{n}, (1 + α(ϵ))g) dominates S^{n}(K + ϵ). Hence G(M) ≤ G(S^{n}(K)) ≤ \sqrt{K}G(S^{n}).

Proof: The required maps f_{ϵ} : M → S^{n}(K + ϵ) are constructed in [3]. □

Corollary 5.5 If M^{n} is a complete Riemannian spin manifold of odd dimension with nonpositive sectional curvature then \frac{π}{r_{0}} 2^{n/2}\sqrt{n} is an upper bound for the length of gaps in the spectrum of the Dirac operator. Here r_{o} := sup_{m∈M} r_{inj}(m). In particular if M has a pole (i.e. M is simply connected) then σ(D) = R.

Corollary 5.6 If M^{n} is a complete Riemannian spin manifold of even dimension with nonpositive sectional curvature then there is spectrum of the Dirac operator D in the interval [−\frac{π}{r_{0}} 2^{\frac{n−3}{2}} \sqrt{n}, \frac{π}{r_{0}} 2^{\frac{n−3}{2}} \sqrt{n}]. If M has a pole then 0 is in the spectrum of D.

In [17] manifolds which dominate spheres of arbitrary large radius are called hyperspherical. In that paper the property 0 ∈ σ(D) was employed as an obstruction against the existence of a metric of positive scalar curvature on hyperspherical manifolds.

6 The number of eigenvalues

In this section we will refine the results of the last section taking into account the actual value of the spectral flow or the index respectively. Let (M^{n}, g) be a Riemannian spin manifold, U ⊂ M be some compact and K(U) := max(sup_{m∈U} K(m) , 0) where K(m) is the maximal sectional curvature at m. We set

r_{0}(U) := min(inf_{p∈U} r_{inj}(p) , \frac{π}{\sqrt{K(U)}})  (49)

and for r ≤ r_{0}(U)

N_{U}(r) := {maximum number of disjoint balls of radius r in U}.  (50)

By the results of [3] there exist maps

f_{ϵ} : M^{n} → S^{n}((π/r)^{2} + ϵ)  (51)

of degree N_{U}(r) with |df| ≤ 1 + α(ϵ) where α(ϵ) → 0 as ϵ → 0. Pulling back gauge transformations and splittings respectively we obtain families of twisted Dirac operators the spectral flow of which is N_{U}(r) in the odd-dimensional case and twisted Dirac operators with index N_{U}(r) in the even-dimensional case. Setting

μ(r) := \frac{π\sqrt{n}}{r} \{ \begin{matrix}2^{n/2} & if n is odd \\ 2^{(n−3)/2} & if n is even \end{matrix} \}  (52)

we have

inf_{ϵ→0} ∥R_{ϵ}∥ ≤ μ(r)  (53)

where R_{ϵ} := R(f_{ϵ}^{∗}W) if n is odd. In the even-dimensional case R_{ϵ} is given by (47) where the splitting is the pull-back by f_{ϵ} of the splitting of the spinor bundle of S^{n} (see [3]).

Theorem 6.1 There are at least \{N_{U}(r)/2^{\{ \frac{n−1}{2} \}}\} eigenvalues of D in the interval [−μ(r), μ(r)] as long as r ≤ r_{0}(U) and μ(r) < inf |σ_{ess}(D)|. Here {x} denotes the smallest integer greater or equal x.

Proof: One applies essentially the same argument as for Theorem 4.1. Note that the dimension of the twisting trivial bundle is dim V = 2^{[ \frac{n−1}{2} ]+1}. □

Define

C(n) := lim inf_{r→0} \frac{\{\frac{N_{U} (r)}{2^{\{ \frac{n−1}{2} \}}} \}}{V ol(U)μ^{n}(r)}  (54)
= \frac{γ_{n}}{V ol(B^{n})n^{n/2}π^{n}} \{ \begin{matrix}2^{n(1−n/2)} & if n is even \\ 2^{\frac{1−n^{2}−n}{2}} & if n is odd \end{matrix} \}  (55)

where γ_{n} is the optimal constant arising in the sphere-packing problem in dimension n and V ol(B^{n}) is the volume of the unit ball in R^{n}.

Corollary 6.2 For every ϵ > 0 there is a T_{0} such that for all T satisfying inf |σ_{ess}(D)| > T ≥ T_{0} holds

♯{λ ∈ σ(D) | |λ| ≤ T } ≥ V ol(U)(C(n) − ϵ)T^{n}.  (56)

A similar result could have been obtained by the Dirichlet-comparison method. But we do not know a reference for explicite values of the constant. This Corollary is interesting since there are examples of Dirac operators on complete Riemannian spin manifolds without essential spectrum [14]. These manifolds may have finite as well as infinite volume. If σ_{ess}(D) = ∅ then

lim inf_{T →0} \frac{♯\{λ ∈ σ(D) | |λ| ≤ T \}}{T^{n}} ≥ V ol(M)C(n).  (57)

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