THE DIRAC OPERATOR AND CONFORMAL COMPACTIFICATION

JOHN LOTT

Abstract. We give results about the L^{2}-kernel and the spectrum of the Dirac operator on a complete Riemannian manifold which admits a conformal compactifation to a compact manifold.

1. Introduction

A general problem in spectral geometry is to understand the spectral properties of Diractype operators on complete Riemannian manifolds. In this paper we show how to use the conformal covariance of the Dirac operator, along with some simple arguments, to derive more precise results than are known for general Dirac-type operators.

For background information about Dirac operators, we refer to [19]. Let M be a connected smooth spin manifold of dimension n > 1. Let S denote the spin bundle of M, equipped with its natural Euclidean inner product.

Let g be a complete Riemannian metric on M, with Dirac operator D_{g}. Let Ker(D_{g}) denote the kernel of D_{g} when acting on L^{2}(S, dvol_{g}). Given σ ∈ C^{∞}(M), let h be the Riemannian metric on M given by

g = e^{2σ} h.  (1.1)

Definition 1. (M, g) has a conformal boundary component if one can find σ as above and a manifold-with-boundary N such that

1. M is diffeomorphic to int(N),

2. ∂N ≠ ∅,

3. h extends to a smooth Riemannian metric on N and

4. e^{− σ} extends to a locally Lipschitz function on N.

If (M, g) has a conformal boundary component then (M, g) is conformally hyperbolic in the sense of [20]. A basic example of a manifold with a conformal boundary component is the real hyperbolic space H^{n}(ℝ), in which case we can take N = \overline{B^{n}} and e^{σ}(x) = \frac{2}{1 − |x|^{2}} for x ∈ B^{n}. In general, we do not require that N be compact.

Theorem 1. If (M, g) has a conformal boundary component then Ker(D_{g}) = 0.

Corollary 1. If (M, g) has a conformal boundary component and 4| dim(M) then zero lies in the spectrum of D_{g}.

Corollary 2. Let Z be a closed connected spin manifold with Â(Z) ≠ 0. Let Γ be a countably infinite discrete group and let Ẑ be a connected normal Γ-cover of Z. If g is a Riemannian metric on Z, let ĝ be the pullback metric on Ẑ. Then (Ẑ, ĝ) does not admit a conformal boundary component.

Date: March 22, 2000.

Research supported by NSF grant DMS-9704633.

Examples :

1. Let M be H^{n}(ℝ). Then M satisfies the conditions of Theorem 1 and Corollary 1. In this case, the conclusions of Theorem 1 and Corollary 1 were previously known by explicit calculation [5, 7].

2. Let M be the complex hyperbolic space H^{2n}(ℂ) = G/K, where G = SU(2n, 1) and K = S(U(2n) × U(1)). There is a Lie algebra representation of s(u(2n) ⊕ u(1)) on R = Λ^{∗}(ℂ^{2n}) given by (m, a) → Λ^{∗}(m) + \frac{a}{2} Id. This integrates to a spinor representation of a double cover K̂ of K on R. Using the isomorphism π_{1}(G) ≅ π_{1}(K), let Ĝ be the corresponding double cover of G. Then H^{2n}(ℂ) = Ĝ/K̂ has a spinor bundle given by S = Ĝ ×K̂ R and a Dirac operator D_{g}. In this case, dim(Ker(D_{g})) = ∞ and zero does not lie in the spectrum of D_{g} [7]. This shows the necessity of the condition in Theorem 1 and Corollary 1 that h be nondegenerate on all of N; note that the numerator in the Poincaré metric

g = \frac{(1 − |z|^{2}) ∑_{i=1}^{2n} dz_{i} ⊗ d\overline{z}_{i} + ∑_{i,j=1}^{2n} \overline{z}_{i}dz_{i} ⊗ z_{j}d\overline{z}_{j}}{(1 − |z|^{2})^{2}}  (1.2)

degenerates on the boundary of the unit disk.

3. As remarked in [4, Pf. of Thm. 6], if m is odd then the manifold M_{n,m} constructed in [10] is an aspherical spin manifold with positive signature, which is the total space of a surface bundle over a surface. Let M̂_{n,m} be the universal cover; it is diffeomorphic to ℝ^{4}. As Â(M_{n,m}) ≠ 0, Corollary 2 implies that for any Riemannian metric g on M_{n,m}, (M̂_{n,m}, ĝ) does not admit a conformal compactification (in the sense of Definition 1) to a 4-disk.

4. The signature operator d + d^{∗} on H^{2n}(ℝ) has an infinite-dimensional L^{2}-kernel [6]. This shows that the analog of Theorem 1 for general Dirac-type operators is false.

Now let (Z, h) be a connected closed Riemannian spin manifold. Let X be a closed subset of Z. We consider complete Riemannian manifolds (M, g) which are conformally equivalent to (Z − X, h). For example, S^{n−m−1} × H^{m+1} is conformally equivalent to S^{n} − S^{m}, where the metric on S^{n−m−1} × H^{m+1} is a product of constant curvature metrics.

Theorem 2. Let (M, g) be a complete connected Riemannian spin manifold of dimension n > 1. Suppose that there is a σ ∈ C^{∞}(M) such that

1. (M, e^{− 2σ}g) is isometrically spin-diffeomorphic to (Z − X, h),

2. σ is bounded below on M and

3. X has finite (n − 2)-dimensional Hausdorff mass.

Then dim(Ker(D_{g})) < ∞.

Corollary 3. In addition to the hypotheses of Theorem 2, suppose that ∫_{M} e^{σ} dvol_{h} < ∞. Then Ker(D_{g}) ≅ Ker(D_{h}).

Corollary 4. Under the hypotheses of Corollary 3, if n is even then the L^{2}-index of D_{g} is ∫_{M} Â(M, g).

Corollary 5. Let (Z, h) be a connected closed even-dimensional Riemannian spin manifold. Let ρ be a nonnegative smooth function on Z whose zero set is a submanifold X ⊂ Z, along which the Hessian of ρ is nondegenerate on the normal bundle T_{X}Z/T_{X}X. Put M = Z −X, with the Riemannian metric g = ρ^{−1} h. Then the L^{2}-index of D_{g} is

Ind_{L^{2}}(D_{g}) = \begin{cases}0 & if dim(X) = dim(Z) − 1, \\ ∫_{M} Â(M, g) & if dim(X) < dim(Z) − 1. \end{cases}  (1.3)

In Section 2 we prove the results stated in the introduction. In Section 3 we make some remarks.

The conformal covariance of the Dirac operator was applied to a different but related problem in [16]. I thank Victor Nistor for pointing this out to me, and for comments on this paper. I thank the Max-Planck-Institut-Bonn for its hospitality while this research was performed.

2. Proofs

If h is as in (1.1), let D_{h} denote the Dirac operator associated to h. As explained in [13, Section 2], the notion of a spinor field on M is independent of the choice of Riemannian metric and so it makes sense to compare D_{g} and D_{h}. From [9, Proposition 1.3] and [12, Proposition 2],

D_{g} = e^{− \frac{(n+1)σ}{2}} D_{h} e^{\frac{(n−1)σ}{2}} .  (2.1)

Let Ker(D_{h}) denote the kernel of D_{h} on L^{2}(S, dvol_{h}). Let Ker_{(σ)}(D_{h}) denote the kernel of D_{h} on L^{2}(S, e^{σ} dvol_{h}). Let M_{e^{\frac{(n−1)σ}{2}}} denote the operator of multiplication by e^{\frac{(n−1)σ}{2}} on sections of S.

Proposition 1. M_{e^{\frac{(n−1)σ}{2}}} is an isometric isomorphism between the Hilbert spaces L^{2}(S, dvol_{g}) and L^{2}(S, e^{σ}dvol_{h}) which restricts to an isometric isomorphism between Ker(D_{g}) and Ker_{(σ)}(D_{h}).

Proof. Given ψ ∈ L^{2}(S, dvol_{g}), put ψ_{σ} = e^{\frac{(n−1)σ}{2}} ψ. Then

∫_{M} |ψ|^{2} dvol_{g} = ∫_{M} |e^{− \frac{(n−1)σ}{2}} ψ_{σ}|^{2} e^{nσ} dvol_{h} = ∫_{M} e^{σ} |ψ_{σ}|^{2} dvol_{h}.  (2.2)

From (2.1),

D_{g} ψ = (e^{− \frac{(n+1)σ}{2}} D_{h} e^{\frac{(n−1)σ}{2}} ) (e^{− \frac{(n−1)σ}{2}} ψ_{σ}) = e^{− \frac{(n+1)σ}{2}} D_{h} ψ_{σ}.  (2.3)

The proposition follows.

Proof of Theorem 1 :

Put N′ = N ∪_{∂N} ((−1, 0] × ∂N), the addition of a collar neighborhood to N. By assumption, h extends to a smooth Riemannian metric h′ on N′. Let D_{h′} be the corresponding Dirac operator on N′.

Given ψ ∈ Ker(D_{g}), put ψ_{σ} = e^{\frac{(n−1)σ}{2}} ψ. From Proposition 1, ψ_{σ} ∈ Ker_{(σ)}(D_{h}). Let ψ′_{σ} be the extension of ψ_{σ} by zero to N′. We claim that ψ′_{σ} is a smooth solution to D_{h′} ψ′_{σ} = 0. Once we show this, it will follow from the unique continuation property of D_{h′}^{2} [1, Theorem on p. 235 and Remark 3 on p. 248], along with the vanishing of ψ′_{σ} on (−1, 0) × ∂N, that ψ′_{σ} = 0 and hence ψ = 0.

To show that ψ′_{σ} is a smooth solution to D_{h′} ψ′_{σ} = 0, by elliptic regularity theory it is enough to show that it is a weak solution to D_{h′} ψ′_{σ} = 0. Let η′_{σ} be a smooth compactlysupported spinor field on N′. Let η_{σ} be the restriction of η′_{σ} to int(N) ≅ M. Then

∫_{N′}⟨D_{h′}η′_{σ}, ψ′_{σ}⟩ dvol_{h′} = ∫_{M}⟨D_{h}η_{σ}, ψ_{σ}⟩ dvol_{h}.  (2.4)

We want to show that this vanishes for each choice of η′_{σ}.

If supp(η′_{σ}) ⊂ int(N) then we can integrate by parts to obtain

∫_{M}⟨D_{h}η_{σ}, ψ_{σ}⟩ dvol_{h} = ∫_{M}⟨η_{σ}, D_{h}ψ_{σ}⟩ dvol_{h} = 0.  (2.5)

Thus we may assume that supp(η′_{σ}) ∩ ∂N ≠ ∅.

Let K be a compact subset of ∂N. For ϵ a small enough positive number, let f : [0, ϵ] × K → N be an embedding given by Fermi coordinates near K. That is, for any k ∈ K, the curve t → f(t, k) is a unit-speed geodesic with f(0, k) = k and (∂_{t}f)(0, k) ⊥ T_{k}∂N. In terms of these coordinates, we can write

h = dt^{2} + m_{t},  (2.6)

where m_{t} is a Riemannian metric on K which depends smoothly on t ∈ [0, ϵ]. To prove that ψ′_{σ} is a weak solution to D_{h′} ψ′_{σ} = 0 we may assume without loss of generality that supp(η_{σ}) ⊂ (0, ϵ] × K for some such K and ϵ. (We are thinking of η_{σ} as being defined on M ≅ int(N). So supp(η_{σ}) is a subset of M which is closed in the topology of M, or equivalently, in the relative topology induced from N. In this sense, (0, ϵ] × K is also a closed subset of M.)

As M is complete, e^{− σ} vanishes on ∂N. Then as e^{− σ} is locally Lipschitz, there is some C > 0 such that when restricted to f ([0, ϵ] × K),

e^{− σ}(t, k) ≤ C t.  (2.7)

For t ∈ (0, ϵ], put K_{t} = f(t, K). As ψ_{σ} ∈ L^{2}(S, e^{σ} dvol_{h}), we have

∫_{0}^{ϵ} C^{−1} t^{−1} ∫_{K_{t}} |ψ_{σ}|^{2} dvol_{m_{t}} dt ≤ ∫_{0}^{ϵ} ∫_{K_{t}} e^{σ} |ψ_{σ}|^{2} dvol_{m_{t}} dt ≤ ∫_{M} e^{σ} |ψ_{σ}|^{2} dvol_{h} < ∞. (2.8)

Thus there is a sequence t_{i} ∈ (0, ϵ] such that lim_{i→∞} t_{i} = 0 and

lim_{i→∞} ∫_{K_{t_{i}}} |ψ_{σ}|^{2} dvol_{m_{t_{i}}} = 0.  (2.9)

As ψ_{σ} ∈ L^{2}(S, e^{σ} dvol_{h}), it follows that the restriction of ψ_{σ} to f((0, ϵ] × K) is squareintegrable with respect to dvol_{h}. Then as D_{h} η_{σ} ∈ L^{2}(S, dvol_{h}), it follows that ⟨D_{h} η_{σ}, ψ_{σ}⟩ ∈ L^{1}(M, dvol_{h}). Given t ∈ (0, ϵ], integration by parts gives

∫_{f([t,ϵ]×K)} ⟨D_{h} η_{σ}, ψ_{σ}⟩ dvol_{h} = ∫_{K_{t}}⟨c(∂_{t}) η_{σ}, ψ_{σ}⟩ dvol_{m_{t}},  (2.10)

where c(∂_{t}) denotes Clifford multiplication by the unit vector ∂_{t}. Then

∫_{M} ⟨D_{h} η_{σ}, ψ_{σ}⟩ dvol_{h} = lim_{i→∞} ∫_{f([t_{i},ϵ]×K)} ⟨D_{h} η_{σ}, ψ_{σ}⟩ dvol_{h}
= lim_{i→∞} ∫_{K_{t_{i}}}⟨c(∂_{t}) η_{σ}, ψ_{σ}⟩ dvol_{m_{t_{i}}}.  (2.11)

Thus

|∫_{M} ⟨D_{h} η_{σ}, ψ_{σ}⟩ dvol_{h}| ≤ lim_{i→∞} |∫_{K_{t_{i}}}⟨c(∂_{t}) η_{σ}, ψ_{σ}⟩ dvol_{m_{t_{i}}} |

≤ lim_{i→∞} (∫_{K_{t_{i}}} |η_{σ}|^{2} dvol_{m_{t_{i}}})^{\frac{1}{2}} (∫_{K_{t_{i}}} |ψ_{σ}|^{2} dvol_{m_{t_{i}}})^{\frac{1}{2}}

≤ ∥ η_{σ} ∥_{∞} sup_{t∈[0,ϵ]} vol^{\frac{1}{2}} (K_{t}) lim_{i→∞} (∫_{K_{t_{i}}} |ψ_{σ}|^{2} dvol_{m_{t_{i}}})^{\frac{1}{2}} = 0.

(2.12)

This proves the theorem.

Proof of Corollary 1 :

Suppose that zero does not lie in the spectrum of D_{g}. Then Theorem 1 implies that D_{g} is a Fredholm operator. Let X be a connected closed n-dimensional spin manifold with Â(X) ≠ 0. Let M′ be the connected sum of M with X. Let K ⊂ M and K′ ⊂ M′ be sufficiently large compact sets and let g′ be a Riemannian metric on M′ for which M′ − K′ is isometric to M − K. Then D_{g′} is also Fredholm. As Theorem 1 applies to M′, we deduce that both D_{g} and D_{g′} have vanishing L^{2}-index. However, by the relative index theorem [8, 18], the difference of the L^{2}-indices is Â(X), which is a contradiction.

Proof of Corollary 2 :

Suppose that (Ẑ, ĝ) admits a conformal boundary component. By Theorem 1, Ker(Dĝ) = 0. However, by Atiyah’s L^{2}-index theorem [2], dim(Ker(Dĝ)) = ∞, which is a contradiction.

Proof of Theorem 2 :

If ψ ∈ Ker(D_{g}), put ψ_{σ} = e^{\frac{(n−1)σ}{2}} ψ. Then ψ_{σ} lies in the kernel of the Dirac operator on Z − X. As ψ_{σ} ∈ L^{2}(S, e^{σ} dvol_{h}) and σ is bounded below, it follows that ψ_{σ} ∈ L^{2}(S, dvol_{h}). From [17, p. 267, 2.3.4], ψ_{σ} extends to an element of Ker(D_{h}) on Z. The unique continuation property of D_{h}^{2} implies that we have constructed a well-defined map from Ker(D_{g}) to Ker(D_{h}), which is clearly injective. As Z is closed, dim(Ker(D_{h})) < ∞.

Proof of Corollary 3 :

In the proof of Theorem 2, we constructed maps Ker(D_{g}) → Ker_{(σ)}(D_{h}) → Ker(D_{h}). By Proposition 1, the first map is an isomorphism. The second map is injective. As ∫_{M} e^{σ} dvol_{h} < ∞, the second map is surjective. The corollary follows.

Proof of Corollary 4 :

From Corollary 3, the L^{2}-index of D_{g} equals the index of D_{h} on Z. From the Atiyah-Singer index theorem, this equals ∫_{Z} Â(Z, h). From the conformal invariance of the Â-form, this in turn equals ∫_{M} Â(M, g).

Proof of Corollary 5 :

Put e^{2σ} = ρ^{−1}.

If dim(X) = dim(Z) − 1, let N be the metric completion of Z − X with respect to h. Then we can apply Theorem 1 to conclude that Ker(D_{g}) = 0.

If dim(X) < dim(Z) − 1 then the claim follows from Corollary 4. We remark that in this case, it follows from [14] that D_{g} has closed range, as in the model case of S^{n} − S^{m}. (When dim(X) = dim(Z) − 2, we are using here the fact that the spin structure on Z − X is inherited from the spin structure on Z.) Hence from Corollary 3, D_{g} is actually Fredholm.

3. Remarks

Any connected complete noncompact Riemannian manifold (M, g) can be conformally compactified to a compact metric space. For example, fix a basepoint m_{0} ∈ M. There are ϕ ∈ C^{∞}(M) and c > 0 such that ϕ(m) ≤ d_{g}(m_{0}, m) ≤ ϕ(m) + c. Taking a function f : ℝ → ℝ which grows sufficiently fast, the metric completion of (M, e^{− f∘ϕ} g) will be the Freudenthal compactification of M, in which one adds a point for each end of M.

In general, even if M has finite topological type, (M, g) will not have a conformal compactification which is a manifold (with or without boundary). The results of this paper suggest that the Dirac operator D_{g} on M can be studied in terms of the possibly-singular spaces which arise as conformal compactifications of M. For example, let (X, g_{X}) be a closed Riemannian spin manifold. Then for any c > 0, an end of M which is isometric to ([0, ∞) × X, dr^{2} + g_{X}) can be conformally compactified to the conical space ([0, 1] × X, ds^{2} + c^{2} s^{2} g_{X}). In this case, the L^{2}-index theorem for manifolds with cylindrical ends [3, 15] says that there will be a contribution to the L^{2}-index formula for D_{g} of − \frac{1}{2} times the eta-invariant of the link of the vertex point of the cone, i.e. of X. This suggests a relationship between Dirac operators on certain complete manifolds and Dirac operators on singular spaces (see [11] and references therein for the latter) although, of course, the relevant L^{2}-spaces are different.

References

[1] N. Aronszajn, “A Unique Continuation Theorem for Solutions of Elliptic Partial Differential Equations or Inequalities of Second Order”, J. Math. Pures Appl. 36, p. 235-249 (1957)

[2] M. Atiyah, “Elliptic Operators, Discrete Groups and von Neumann Algebras”, Astérisque 32, p. 43-72 (1976)

[3] M. Atiyah, V. Patodi and I. Singer, “Spectral Asymmetry and Riemannian Geometry I”, Math. Proc. Cambridge Phil. Soc. 77, p. 43-69 (1975)

[4] C. Bohr, “On the Signatures of Even 4-Manifolds”, preprint, http://xxx.lanl.gov/abs/math.GT/0002151 (2000)

[5] U. Bunke, “The Spectrum of the Dirac Operator on the Hyperbolic Space”, Math. Nachr. 153, p. 179-190 [6] H. Donnelly, “The Differential Form Spectrum of Hyperbolic Space”, Manuscripta Math. 33, p. 365-385

(1981)

[7] S. Goette and U. Semmelmann, “The Point Spectrum of the Dirac Operator on Noncompact Symmetric Spaces”, preprint, http://xxx.lanl.gov/abs/math.DG/9903177 (1999)

[8] M. Gromov and H.B. Lawson, “Positive Scalar Curvature and the Dirac Operator on Complete Riemannian Manifolds”, Publ. Math. IHES 58, p. 83-196 (1983)

[9] N. Hitchin, “Harmonic Spinors”, Advances in Mathematics 14, p. 1-55 (1974)

[10] K. Kodaira, “A Certain Type of Irregular Algebraic Surfaces”, J. d’Analyse Math. 19, p. 207-215 (1967)

[11] M. Lesch, \underline{Operators of Fuchs Type, Conical Singularities and Asymptotic Methods}, Teubner Texts in Mathematics 136, Teubner-Verlag, Stuttgart (1997)

[12] J. Lott, “Eigenvalue Bounds for the Dirac Operator”, Pac. J. Math. 125, p. 117-126 (1986)

[13] J. Lott, “Â-Genus and Collapsing”, preprint at http://www.math.lsa.umich.edu/ lott, to appear, J. of Geom. Anal. (1997)

[14] R. Mazzeo, “Elliptic Theory of Differential Edge Operators I”, Comm. Part. Diff. Eq. 16, p. 1615-1664 (1991)

[15] W. Müller, “L^{2}-Index and Resonances”, in \underline{Geometry and Analysis on Manifolds}, Lect. Notes in Math. 1339, Springer-Verlag, New York (1988)

[16] V. Nistor, “On the Kernel of the Equivariant Dirac Operator”, Ann. Global Anal. Geom. 17, p. 595-613 (1999)

[17] J. Polking, “A Survey of Removable Singularities”, in \underline{Seminar on Nonlinear Partial Differential Equations}, ed. S.S. Chern, MSRI Publications 2, Springer-Verlag, New York, p. 261-292 (1984)

[18] J. Roe, “A Note on the Relative Index Theorem”, Quart. J. Math. Oxford Ser. 42, p. 365-373 (1991)

[19] J. Roe, \underline{Elliptic Operators, Topology and Asymptotic Methods}, Second Edition, Pitman Research Notes in Mathematics Series, 395, Longman, Harlow (1998)

[20] V. Zorich, “On the Conformal Type of a Riemannian Manifold”, Funct. Anal. Appl. 30, p. 106-117 (1996)

Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1109, USA

  E-mail address: lott@math.lsa.umich.edu