HODGE THEORY AND COHOMOLOGY

WITH COMPACT SUPPORTS

Ed Bueler Igor Prokhorenkov

April 27, 1998

Abstract. This paper constructs a Hodge theory of noncompact topologically tame manifolds M. The main result is an isomorphism between the de Rham cohomology with compact supports of M and the kernel of the Hodge–Witten–Bismut Laplacian △_{μ} associated to a measure dμ which has sufficiently rapid growth at infinity on M. This follows from the construction of a space of forms associated to △_{μ} which satisfy an “extension by zero” property. The “extension by zero” property is proved for manifolds with cylindrical ends possessing gaussian growth measures.

1. Introduction

Let M be a real C^{∞} manifold without boundary. We say M is topologically tame if there exists a compact manifold N with boundary such that M is diffeomorphic to the interior of N. For instance, vector bundles over compact base manifolds are topologically tame. Compact manifolds are trivially topologically tame. A surface of infinite genus is not topologically tame.

We will construct a Hodge theory of topologically tame noncompact manifolds M in this paper. First, we choose a metric g such that (M, g) is complete and the ends of M are metric cylinders. Then we define a particular (Hodge–Witten– Bismut) Laplacian operator △_{μ} on certain differential forms on M. The subscript μ denotes an additional choice of measure dμ. The operator △_{μ} acts on forms which are square–integrable with respect to the measure dμ.

The essential construct of this paper is that the measure dμ has certain rapid growth at infinity on M relative to Riemann–Lebesgue measure dx: dμ = e^{cr^{2}}dx, c > 0. In this paper we prove (theorem 15):

Main Theorem. With these choices, (M^{n}, g), dμ, the △_{μ}–harmonic forms represent the de Rham cohomology with compact supports:

ker △_{μ}^{p} ≅ H_{c,de R}^{p}(M), p = 0, …, n.

Research at MSRI is supported in part by NSF grant DMS–9701755.

The skeptical reader may observe that any nonzero ω ∈ ker △_{μ}, as an eigenform of an elliptic operator, does not have compact support. Nevertheless, we will exhibit a (not canonical!) linear surjection j : ker △_{μ} → H_{c,de R}(M) and a (also not canonical) linear surjection Π : H_{c,de R}(M) → ker △_{μ}, in the situation covered by the main theorem.

In fact, we will consider a condition on more general measures dμ and on the domain of their associated operators △_{μ}. This condition is the extension by zero condition “property EZ” defined in the third section, and it is the main technique used in proving theorem 15:

Main Technique. Choose a “nice” relatively compact open submanifold M_{0} ⊂ M such that M_{0} is diffeomorphic to M under Ψ : M → M_{0}. Show that if ω ∈ C^{∞}(△_{μ}) = ⋂_{k≤1} D_{(△_{μ})^{k}} then (Ψ^{−1})^{∗} ω ∈ Ω(M_{0}) has smooth extension by zero to M. (“Property EZ”) Show that this condition implies the existence of a surjection j : ker △_{μ} → H_{c,de R}(M).

Note that any eigenforms of △_{μ} are in C^{∞}(△_{μ}), and that they decay very rapidly if dμ = e^{cr^{2}}dx, c > 0 as prescribed.

Property EZ can hold only if M is topologically tame.

Theorem 13 is that property EZ implies the existence and surjectivity of j. The remainder of the introduction defines △_{μ} and motivates our choice of dμ.

We also recall the existing Hodge theory of △, constructed in the usual way from dx, in the case of (M, g) with cylindrical ends. Comparison then shows the role of the densities e^{±r^{2}} from the point of view of the Hodge theory of such (M, g).

Now, the metric g determines an inner product (·, ·)_{x} on the tangent space T_{x}M by definition, which induces an inner product (·, ·)_{x} on ∧T_{x}^{∗}M, the space of pointwise values of differential forms. The metric also determines a measure dx, by the coordinate formula dx = \sqrt{det g_{ij}}dx^{1} … dx^{n}.

Assume M is oriented. It is traditional to define an inner product on the space of smooth (C^{∞}) compactly–supported differential forms Ω_{c}(M) by the formula ⟨ω, ν⟩ = ∫_{M} (ω, ν)_{x} dx = ∫_{M} ω ∧ ⋆ν. (See [Wa], for instance, for details.) Here ⋆ is the Hodge duality operator defined in terms of (·, ·)_{x} and the orientation on M.

Next we weight the inner product. Restricting ourselves to smooth, positive densities, if h ∈ C^{∞}(M) then define

dμ = e^{2h}dx  and  ⟨ω, ν⟩_{μ} = ∫_{M} (ω, ν)_{x} dμ.

Let L_{μ}^{2}Ω be the completion of Ω_{c}(M) with respect to ⟨·, ·⟩_{μ}. The exterior derivative d has formal adjoint δ_{μ}: ⟨dω, ν⟩_{μ} = ⟨ω, δ_{μ}ν⟩_{μ} for ω, ν ∈ Ω_{c}. In fact, the formula δ_{μ} = e^{−2h}δe^{2h} holds, where δ = (−1)^{n(p+1)+1} ⋆ d⋆ acting on Ω^{p} is the formal adjoint of d with respect to the usual inner product.

Let △_{μ} = dδ_{μ} + δ_{μ}d. It is well–known that

1. Theorem. The operator △_{μ} defined on Ω_{c} is densely–defined and symmetric with respect to ⟨·, ·⟩_{μ}, and is a second–order elliptic operator which maps Ω_{c}^{p} to itself. If (M, g) is complete, we can extend △_{μ} to a unique nonnegative (self– adjoint) operator in L_{μ}^{2}Ω ([G], [C]—see [Bu] for the weighted case).

If M^{n} is in fact compact, then △_{μ} has compact resolvent (discrete spectrum). The Hodge decomposition of the smooth forms Ω^{p}(M), p = 0, …, n, follows:

Ω^{p}(M) = ker △_{μ}^{p} ⊕ im d^{p−1} ⊕ im δ_{μ}^{p+1}, (M compact)

from which it follows that H_{de R}^{p}(M) ≅ ker △_{μ}^{p}.

The weighting dμ = e^{2h}dx on L^{2} spaces of forms has been considered in connection with the Witten treatment of the Morse inequalities [Wi]. Actually, our △_{μ} on L_{μ}^{2}Ω is unitarily–equivalent to the Laplacian usually considered in that context, as explicitly noted in [Bi]. There is the following unitary equivalence:

(1)  \begin{matrix}L_{μ}^{2}Ω & ⟶^{U} & L^{2}Ω \\ ω & ↦ & e^{h}ω \\ d & & d_{h} = UdU^{−1} = e^{h}de^{−h} \\ △_{μ} = dδ_{μ} + δ_{μ}d & & △_{h} = U△_{μ}U^{−1} = △ + |dh|^{2} + A_{h}, \end{matrix}

where A_{h} is a certain zero order symmetric operator on L^{2}Ω(M) involving the second derivatives of h.

We want to consider noncompact topologically tame manifolds M. On such M we can explicitly construct metric and measure which yield nice spectral properties.

2. Theorem. ([Bu], Theorem 10.2) Let M be a C^{∞}, orientable, connected, and topologically tame manifold. There exists a metric g and for each c ≠ 0 a measure dμ with smooth density with respect to dx such that

(i) (M, g) is complete and
(ii) △_{μ} in L_{μ}^{2}Ω has compact resolvent (= discrete spectrum).

Under the chosen metric g, the ends of M are cylinders near infinity. The density of the chosen measure dμ behaves precisely as exp(cr^{2}) near infinity on the ends, for any chosen c ≠ 0.

A certain Hodge decomposition is one consequence of (ii). It is sometimes misnamed a strong Hodge decomposition. It says:

3. Proposition. If (M^{n}, g) is complete and △_{μ} has compact resolvent, or merely a spectral gap (i.e. σ(△_{μ}) \ {0} ⊆ [ϵ, ∞) for some ϵ > 0), then

L_{μ}^{2}Ω^{p} = ker △_{μ}^{p} ⊕ im d^{p−1} ⊕ im δ_{μ}^{p+1}, p = 0, …, n.

Proof. Classical. Use the spectral theorem. See [Bu] for details. □

The conclusion of proposition 3 is not strong in the sense that there is a priori no relation between ker △_{μ} and either H_{de R}(M) or H_{c,de R}(M) in the noncompact case. On must produce a separate proof of any correspondence. We will do so in the case covered by theorem 2—we will show that if c > 0 then ker △_{μ} ≅ H_{c,de R}(M) and if c < 0 then ker △_{μ} ≅ H_{de R}(M).

The widely studied case dμ = dx provides plenty of examples showing that ker △ and H_{de R}(M) at best stand in a very complicated relation to each other. The surveys [An] and [L] should serve as adequate illustration.

A positive example in the case dμ = dx is that if M has the additional structure of a Riemannian cover of a compact quotient manifold M/Γ, then the space ker △ is determined by the topology of M/Γ and by the group Γ. This situation was addressed by Atiyah [At]. (It should be clear that in general these noncompact M, with compact quotient M/Γ, need not be topologically tame. Conversely, many topologically tame noncompact M admit no metric under which there exists a group of isometries Γ with compact quotient M/Γ.)

The class of Riemannian manifolds (M, g) we consider in theorems 2 and 15— topologically tame and with cylindrical ends—has been a subject of Hodge–theoretic consideration. Again take dμ = dx. The following theorem of Atiyah, Patodi, and Singer applies:

4. Theorem. ([APS], Proposition 4.9) Assume (M, g) is complete, is topologically tame and has cylindrical ends. Let i : Ω_{c}(M) → Ω(M) be the inclusion map. The space ker △ of L^{2} harmonic forms is isomorphic to the image of the induced map i from H_{c,de R}(M) into H_{de R}(M).

Although this result establishes a connection between the space of L^{2} harmonic forms and the (topological) cohomologies, it shows that ker △ must vanish in circumstances where there is much cohomology to be represented. For instance, ker △^{p} vanishes for all 0 ≤ p ≤ n if M^{n} is not compact and either H_{de R}^{p}(M) = 0 for 1 ≤ p ≤ n/2 or H_{de R}^{p}(M) = 0 for n/2 ≤ p ≤ n − 1. (Recall that ker △^{p} ≅ ker △^{n−p} for any (M, g) and that H_{de R}^{n}(M) = 0 if M is not compact.)

Theorem 4 and the paper [APS] have motivated a family of results on manifolds with cylindrical ends. R. Melrose and coworkers (specifically, [M]) address the situation of a compact manifold with boundary, and a certain prescribed metric near the boundary which they call an “exact b–metric”. Section 6.4 of [M] includes a Hodge theorem which may be equivalent to our theorem 15 after mapping the manifold with boundary and exact b–metric to a noncompact manifold with cylindrical ends. However, the technique given in section 3 of this paper applies to manifolds with noncylindrical ends, and furthermore directly addresses the complete noncompact manifold itself. For instance, the “boundary condition” here, that makes △_{μ} self–adjoint, is exactly the condition that (M, g) is complete.

Now, we are motivated to consider measures with particular decay rates by the following example, based on well–known spectral properties of the quantum harmonic oscillator.

5. Example. Let dμ^{+} = e^{+|x|^{2}}dx, dμ^{−} = e^{−|x|^{2}}dx on Euclidean space R^{n}. Then △_{μ^{+}} and △_{μ}^{−} have compact resolvent and

ker △_{μ^{−}}^{p} ≅ \{ \begin{matrix}R, & p = 0 \\ 0, & p > 0, \end{matrix} ker △_{μ^{+}}^{p} ≅ \{ \begin{matrix}0, & p < n \\ R, & p = n. \end{matrix}

Note that

H_{de R}^{p}(R^{n}) ≅ \{ \begin{matrix}R, & p = 0 \\ 0, & p > 0, \end{matrix} H_{c,de R}^{p}(R^{n}) ≅ \{ \begin{matrix}0, & p < n \\ R, & p = n. \end{matrix}

In the example above, it is clear that the cases dμ^{+} = e^{c|x|^{2}}dx for c > 0 and dμ^{−} = e^{c|x|^{2}}dx for c < 0 on R^{n} differ fundamentally, but we can easily show their close relation as well. In general, if dμ^{+} = e^{+2h}dx and dμ^{−} = e^{−2h}dx on M, then from (1)

△_{+h} = U_{+}△_{μ^{+}}U_{+}^{−1} = △ + |dh|^{2} + A_{h},
△_{−h} = U_{−}△_{μ^{−}}U_{−}^{−1} = △ + |dh|^{2} − A_{h},

are the unitarily equivalent versions of △_{μ^{+}}, △_{μ^{−}} respectively, both acting on (unweighted) L^{2}Ω. Thus the difference between △_{μ^{+}} and △_{μ^{−}} can be thought of as determined only by the sign of the potential term A_{h} (an endomorphism on ∧T^{∗}M), once we push △_{μ^{+}}, △_{μ^{−}} back into the same space L^{2}Ω. On the other hand, note that L_{μ^{+}}^{2}Ω and L_{μ^{−}}^{2}Ω contain forms of (in general) very different growth/decay at infinity.

In the rest of this paper, we will drop “de R” from our notation, but it is appropriate to note that our results relating analytic quantities to topological quantities are “modulo de Rham’s theorem”. Recall that de Rham’s theorem identifies the de Rham space ker d/ im d defined in the C^{∞} category with the real cohomology space H(M; R) defined in the continuous category.

Now, if ker △_{μ}^{p} represents H^{p}(M) for all 0 ≤ p ≤ n for a given noncompact M, then it will not also represent H_{c}^{p}(M). However, as suggested by example 4, measures with inverse densities give a desirable relationship.

6. Proposition. Poincaré duality. If M^{n} is orientable, then

H^{p}(M) ≅ (H_{c}^{n−p}(M))^{∗} .

Weighted Poincaré duality. If (M^{n}, g) is orientable, and dμ^{+} = e^{+2h}dx, dμ^{−} = e^{−2h}dx, then

ker △_{μ^{−}}^{p} ≅ ker △_{μ^{+}}^{n−p}.

Proof. For Poincaré duality, see [BT]. For this weighted version, see [Bu]. □

It follows that if we find a measure dμ = e^{2h}dx for which ker △_{μ}^{p} ≅ H_{c}^{p}(M), then defining dμ^{−} = e^{−2h}dx, we get (for free!) ker △_{μ^{−}}^{p} ≅ H^{p}(M). Theorem 15 shows that this is the situation that holds for a topologically tame manifold, having chosen g and dμ for c > 0 and c < 0, respectively, as in theorem 2.

The methods of this paper are closely related to those in [P], where, in particular, the isomorphism between the L^{2} cohomology of a vector bundle E, corresponding to a density e^{c|x|^{2}} for c > 0, and the compact cohomology H_{c}(E) was proven, and then used in a proof of the Morse–Bott inequalities.

However, the particular form of definitions 7, 9 and 11 and of theorem 13 are motivated by the following situation. If (M, g) is complete and has Ricci curvature bounded below, then the usual Laplace–Beltrami operator −\frac{1}{2}△ on M has a unique heat kernel ρ. On Euclidean space R^{n}, ρ_{t}(x, y) = (2πt)^{−n/2} exp(−|x − y|^{2}/2t). On M with Ricci bounded below, if t > 0 and x_{0} ∈ M then dμ^{−} ≡ ρ_{t}(x_{0}, x)dx is a probability measure on M. It is conjectured in [Bu] that ker △_{μ^{−}} ≅ H^{p}(M) for this heat kernel measure dμ^{−}, and certain evidence is given there. It is hoped that the methods of this paper will apply to dμ^{+} ≡ (ρ_{t}(x_{0}, x))^{−1} dx, in which case the conjecture would hold true. The conjecture implies an existence conjecture: if b_{p} > 0 then △_{μ^{−}}^{p} has nonvanishing kernel.

We thank Rostislav Matveyev for conversations and help with lemma 12. The second author thanks Maxim Braverman for valuable disscussions about the subject of section 5.

2. Forms of Very Rapid Decay C_{μ}^{∞}

7. Definition. Assume (M, g) is complete, so △_{μ} is self–adjoint. Define

C_{μ}^{∞} ≡ C^{∞}(△_{μ}) = ⋂_{k≥1} D_{(△_{μ})^{k}}.

Clearly Ω_{c}(M) ⊆ C_{μ}^{∞}, so C_{μ}^{∞} is never empty. If △_{μ} has eigenforms, they too are in C_{μ}^{∞}: △_{μ}ω = λω clearly implies ω ∈ ⋂_{k} D_{(△_{μ})^{k}}.

The ellipticity of △_{μ} implies that C_{μ}^{∞} ⊆ Ω(M), i.e. that C_{μ}^{∞} forms are smooth. If M is compact, then C_{μ}^{∞} = Ω(M). If M is not compact, then C_{μ}^{∞} includes constraints on growth at infinity, and in fact Ω_{c}(M) ⊂ C_{μ}^{∞} ⊂ Ω(M) are both proper inclusions.

It is appropriate to think of C_{μ}^{∞} as some kind of generalization to forms of the Schwartz space of functions, with the growth bounds determined by dμ. More accurately, e^{h}C_{μ}^{∞} = C^{∞}(△_{h}) generalizes Schwartz space—see section V.

Growth constraints are included in definition 7. Consider the example:

8. Example. Let M = R^{n} with the Euclidean metric, and let dμ = e^{+x^{2}}dx. The unitarily equivalent operator △_{h} on L^{2}Ω, corresponding to △_{μ} on L_{μ}^{2}Ω, in this case is the harmonic oscillator Hamiltonian △_{h}^{p} = △^{p} + |x|^{2} + n − 2p (on L^{2}Ω^{p}).

Considering only functions, it is well known that C^{∞}(△_{h}^{0}) corresponds to the Schwarz space S(R^{n}). In fact, the family of seminorms {∥f∥_{k} = ∥(△_{h}^{0})^{k}f∥_{L^{2}}} is equivalent to the usual family of seminorms {∥f∥_{α,β} = ∥x^{α}D^{β}f∥_{L^{2}}} ([RS], Appendix to V.3). If follows that f ∈ C^{∞}(△_{h}^{0}) iff ∥f∥_{k} < ∞ for all k iff f ∈ S(R^{n}).

Therefore g ∈ C_{μ}^{∞} iff g = e^{−|x|^{2}/2}f for f ∈ S(R^{n}). This shows that functions in C_{μ}^{∞} have “very rapid” decay.

9. Definition. H_{μ} is the cohomology of (C_{μ}^{∞}, d):

H_{μ}^{p} ≡ \frac{ker d|_{C_{μ}^{∞,p}}}{im d|_{C_{μ}^{∞,p−1}}} , p = 0, …, n.

The closure of im d|_{C_{μ}^{∞}} is not taken in our definition of H_{μ}. Therefore H_{μ} ≅ ker △_{μ} if △_{μ} has a spectral gap:

10. Corollary. The Hodge decomposition of proposition 3 can be smoothed: If △_{μ} has a spectral gap, then

C_{μ}^{∞} = ker △_{μ} ⊕ im d|_{C_{μ}^{∞}} ⊕ im δ_{μ}|_{C_{μ}^{∞}}

so ker △_{μ} ≅ H_{μ}.

Proof. First, clearly ker △_{μ} ⊂ C_{μ}^{∞}. Thus from proposition 3,

C_{μ}^{∞} = ker △_{μ} ⊕ C_{μ}^{∞} ∩ (im d ⊕ im δ_{μ}) .

Suppose ω ∈ C_{μ}^{∞}, ω = dϕ + δ_{μ}ψ, with ϕ,ψ not necessarily smooth. Recall that if ν ∈ ker △_{μ} then dν = δ_{μ}ν = 0. We have

⟨ν, ω⟩_{μ} = ⟨ν, dϕ⟩_{μ} + ⟨ν, δ_{μ}ψ⟩_{μ} = ⟨δ_{μ}ν, ϕ⟩_{μ} + ⟨dν, ψ⟩_{μ} = 0

for all ν ∈ ker △_{μ}. So ω ∈ (ker △_{μ})^{⊥}. Since △_{μ} has a spectral gap, (△_{μ})^{−1} is a bounded operator on (ker △_{μ})^{⊥}. Thus ω = △_{μ}△_{μ}^{−1}ω = d (δ_{μ}△_{μ}^{−1}ω)+δ_{μ} (d△_{μ}^{−1}ω). That is, we can take ϕ = δ_{μ}△_{μ}^{−1}ω, ψ = d△_{μ}^{−1}ω in C_{μ}^{∞}. □

3. Extension–by–Zero and Compactly–Supported Cohomology

We start with an obvious construction on a topologically tame manifold.

Suppose M is topologically tame and is equipped with a metric g making (M, g) a complete metric space. Then we can think of M as including a finite diameter part carrying all the interesting topology, and a finite list of infinite–length ends which are topological products. In fact, let Φ : int N → M be a diffeomorphism. Let Û ⊂ N be any open neighborhood of ∂N which is small enough so that Û ≅ ∂N × (0, ∞]. The closure of Û in N is compact. Decompose Û into finitely–many connected pieces, so Û = ⨆_{i} Q_{i} × (0, ∞], with each Q_{i} compact without boundary. The boundary ∂N is identified with points (q ∈ Q_{i}, r = ∞) in this representation of Û. Define U = ⨆_{i} Q_{i} × (0, ∞). Let M_{0} = M \ \overline{Φ(U)}, so M_{0} has compact closure in M and finite diameter, and let E_{i} = Φ(Q_{i} × (0, ∞)). We can decompose M = \overline{M_{0}} ⊔ (⨆_{i} E_{i}). Note that M_{0} is diffeomorphic to M itself.

Now we consider the induced mapping (Ψ^{−1})^{∗}, for Ψ : M → M_{0} a diffeomorphism, taking “very–rapid–decay” forms C_{μ}^{∞} back to M_{0}.

11. Definition. Property EZ. Let M be topologically tame, assume (M, g) is complete, and suppose a measure dμ = e^{2h}dx is given. Consider M_{0} ⊂ M an open submanifold of M with compact closure, such that ∂M_{0} ⊂ M is a smooth closed submanifold, and such that there exists a diffeomorphism Ψ : M → M_{0}. Such M_{0} exist since M is topologically tame. We say dμ has Property EZ if there exists M_{0} and Ψ so that every ω ∈ (Ψ^{−1})^{∗}C_{μ}^{∞} has a smooth extension by zero to all of M.

From now on, we will write H_{c}(M) for the cohomology of the complex (Ω_{c}(M), d), i.e. for the de Rham cohomology with compact supports. Note also that the cohomology spaces H_{c}(M_{0}), H_{c}(M) in the following lemma are finite–dimensional because M is topologically tame.

12. Lemma. Let M be a topologically tame manifold. Suppose M_{0} ⊂ M is an open submanifold with compact closure, Ψ : M → M_{0} a diffeomorphism. Assume ∂M_{0} is a smooth (closed) submanifold of M. Let i : Ω_{c}(M_{0}) → Ω_{c}(M) be the map which extends forms by zero. Then i induces an isomorphism i : H_{c}(M_{0}) → H_{c}(M).

Proof. We construct a family Ψ_{ϵ}, 0 < ϵ < 1, of M → M_{0} diffeomorphisms which fix “large portions” of M_{0}. First, note that Ψ(M_{0}) is diffeomorphic to M, since Ψ|_{M_{0}}∘Ψ is a diffeomorphism. Thus it is clear that M \Ψ(M_{0}) is diffeomorphic to ∂M_{0} times an interval. Let α : ∂M_{0} × [−1, ∞) → M \ Ψ(M_{0}) be such a diffeomorphism, chosen so that α(∂M_{0} × [−1, 0)) = M_{0} \ Ψ(M_{0}) and α(∂M_{0} × [0, ∞)) = M \ M_{0}, so α(∂M_{0} × {0}) = ∂M_{0}. Let M_{ϵ} = M_{0} \ α(∂M_{0} × (−ϵ, ∞)) for 0 < ϵ < 1, and also define β_{ϵ} : [−1, ∞) → [−1, 0) so that β_{ϵ}(ξ) = ξ for −1 ≤ ξ ≤ −ϵ, and β_{ϵ} is smooth and monotone increasing with lim_{ξ→∞} β_{ϵ}(ξ) = 0.

Define Ψ_{ϵ}(x) = x for x ∈ Ψ(M_{0}). For x ∈ M \ Ψ(M_{0}), let (y, ξ) = α^{−1}(x), y ∈ ∂M_{0}, ξ ∈ [−1, ∞). Define Ψ_{ϵ}(x) = α(y, β_{ϵ}(ξ)). Then Ψ_{ϵ} fixes M_{ϵ} and is a diffeomorphism M → M_{0} for each 0 < ϵ < 1.

Now, Ψ_{ϵ} induces Ψ_{ϵ}^{∗} : Ω_{c}(M_{0}) → Ω_{c}(M) on forms and an isomorphism Ψ_{ϵ}^{∗} : H_{c}(M_{0}) → H_{c}(M).

If α ∈ H_{c}(M_{0}), α ≠ 0, we will show iα ≠ 0 in H_{c}(M). This suffices, since H_{c}(M_{0}), H_{c}(M) are finite–dimensional. Choose any representative ω, so [ω] = α. Since ω has compact support in M_{0}, there exists ϵ > 0 such that supp ω ⊂ M_{ϵ}. Then Ψ_{ϵ}^{∗}ω = iω. But iα = [iω] = [Ψ_{ϵ}^{∗}ω] = Ψ_{ϵ}^{∗}α ≠ 0 in H_{c}(M) since Ψ_{ϵ}^{∗} is an isomorphism. □

13. Theorem. Suppose M^{n} is topologically tame, assume (M, g) is complete, and suppose the measure dμ = e^{2h}dx is given. If dμ has property EZ, then there exists a map j : C_{μ}^{∞} → Ω_{c}(M), which restricts to a surjection

j : H_{μ} → H_{c}(M).

Proof. Let M_{0}, Ψ be as in the definition of “property EZ”. Define the map j to take ω ∈ C_{μ}^{∞} to the extension by zero of (Ψ^{−1})^{∗}ω. Let k : Ω_{c}(M_{0}) → C_{μ}^{∞} be extension by zero. Then i factors through j: i = j ∘ k.

By the lemma, i is an isomorphism in cohomology. At the level of cohomology,

H_{c}(M_{0}) ⟶^{k} H_{μ} ⟶^{j} H_{c}(M), i = j ∘ k isomorphism,

it follows that k is an injection and j is a surjection. □

Recalling Poincaré duality, we see that if dμ on (M^{n}, g) has property EZ then

dim H_{μ}^{p} ≥ dim H_{c}^{p}(M) = dim H^{n−p}(M) ≡ b_{n−p}.

We do not exclude the possibility that dim H_{μ}^{p} = ∞, nor do we assume that H_{μ}^{p} has a Hilbert space structure. Also, it is not assumed that △_{μ} has a spectral gap.

There is an available map going the other direction from j. Consider the orthogonal projection Π : L_{μ}^{2}Ω → ker △_{μ}. This map clearly restricts to Π : Ω_{c}(M) → ker △_{μ}. We also use Π to denote the map on compactly–supported cohomology:

14. Definition. Assume (M, g) is complete. Define

Π : H_{c}(M) → ker △_{μ}  by  [ω] ↦ Π(ω).

Recall that if M is complete, then ⟨dν, φ⟩_{μ} = ⟨ν, δ_{μ}φ⟩_{μ} = 0 for all ν ∈ Ω_{c}(M) and φ ∈ ker △_{μ}. This shows that Π is well–defined.

If dμ has property EZ, then the following method may also apply. First, the natural map η : ker △_{μ} → H_{μ}, given by ω ↦ [ω] is always injective if M is complete, since if ω = dν for ν ∈ C_{μ}^{∞} then ⟨ω, φ⟩_{μ} = ⟨ν, δ_{μ}φ⟩_{μ} = 0 for all φ ∈ ker △_{μ}. If dμ has property EZ, and if the composition

ker △_{μ} ⟶^{η} H_{μ} ⟶^{j} H_{c}(M) ⟶^{Π} ker △_{μ}

is surjective, then Π is surjective. This reduces to checking whether an explicit endomorphism Π ∘ j ∘ η on ker △_{μ}, depending only on Ψ and dμ, is surjective.

Whether Π is surjective or not is a priori independent of the existence of a spectral gap for △_{μ}.

In section 4 we will see that both j and Π are surjective in the case described by theorem 2 if c > 0, and thus that H_{μ} ≅ H_{c}(M). That is, we prove that if: (i) M is topologically tame, (ii) the ends of (M, g) are cylinders, and (iii) dμ is a measure with density exp(cr^{2}), c > 0 on the ends, then dμ has property EZ. It follows that j is surjective, and we prove Π is surjective by the method described above.

4. Cylindrical Ends, Gaussian Growth Measure

Let us return to the setting of theorem 2. Let M be topologically tame, and define g as a metric under which (M, g) is complete and the ends of M are cylinders. Define dμ as in theorem 2 as well, with c > 0: dμ = e^{cr^{2}}dx. (We will more explicitly define g and dμ in a moment.) In this section we prove that dμ on M has property EZ, and in fact we prove that both j and Π of the previous section are surjections. As we will see, the proof of property EZ ultimately depends on a fundamental estimate for C_{μ}^{∞} (actually C_{h}^{∞}) forms on (M, g). The estimate is found in theorem 16 and corollary 17 of section 5.

15. Theorem. Let c > 0. Let M, g, dμ = e^{cr^{2}}dx be as in theorem 2. Then in addition to (i) and (ii), there exist surjections Π and j such that

H_{c}(M) ⟶^{Π} ker △_{μ} ≅ H_{μ} ⟶^{j} H_{c}(M).

It follows that

H_{μ} ≅ H_{c}(M).

Proof that j is surjective. We show that dμ has property EZ, so that by theorem 13, j : H_{μ} → H_{c}(M) is a surjection.

Since M is topologically tame, M = M_{0} ∪ (⨆_{i} E_{i}), with M_{0} relatively compact, E_{i} = (0, ∞) × Q_{i}, and Q_{i} compact without boundary. Let g be a product metric on each E_{i} and otherwise choose g arbitrarily.

Fix an end E = (0, ∞) × Q, and let Q = ⋃_{j} U_{j} be a finite cover of Q by coordinate neighborhoods. Let (x^{1}, x^{2}, …, x^{n}) = (x^{1}, x̄) = (r, x̄) be coordinates on (0, ∞) × U_{j}.

Define dμ = e^{cr^{2}}dx = e^{c(x^{1})^{2}}dx on (0, ∞) × U_{j}, therefore on E. Extend dμ to all of M in the obvious way, so dμ = e^{+2h}dx on M, where h ∈ C^{∞}(M) and h = cr^{2} on each end, for the given c > 0.

Define Ψ on E as follows. Let M_{1} = M_{0} ∪ (⨆_{i}(0, 1) × Q_{i}), with coordinates (s, ȳ) on (0, 1) × U_{j}. We define Ψ^{−1} : M_{1} → M by defining Ψ^{−1} = id on M_{0} and by defining Ψ^{−1} : (0, 1) × U_{j} → (0, 1) × U_{j} as follows: Ψ^{−1}(s, ȳ) = ((1 − s)^{−1}, ȳ) if 1/2 ≤ s < 1, and smoothly and monotonically interpolate Ψ^{−1} for 0 ≤ s ≤ 1/2. Clearly Ψ^{−1} is a diffeomorphism and M_{1} = Ψ(M) has compact closure.

To prove that dμ has property EZ, it suffices to prove that if ω ∈ C_{μ}^{∞} and supp ω ⊂ E, then (Ψ^{−1})^{∗} ω has smooth extension by zero to E. That is, if φ_{0},{φ_{i}} is a partition of unity subordinate to M = M_{1} ∪(⨆_{i} E_{i}) then ω ∈ C_{μ}^{∞} iff φ_{i}ω ∈ C_{μ}^{∞} for each i.

Now we arrange to use the properties of Schwartz forms on an end E—see section V—to prove the extension by zero on E for (Ψ^{−1})^{∗} ω.

In fact, recall that △_{h} = e^{h}△_{μ}e^{−h} = △ + |dh|^{2} + A_{h} on L^{2}Ω is unitarily– equivalent to △_{μ} on L_{μ}^{2}Ω. Here h = \frac{c}{2}r^{2}. Let ν ∈ C_{h}^{∞} = ⋂_{k≥1} D_{(△_{h})^{k}}. Then e^{−h}ν ∈ C_{μ}^{∞}. Suppose ν = ∑_{1∉J} u_{J}dr ∧ dx̄^{J} + ∑_{1∉I} u_{I}dx̄^{I} on (0, ∞) × U_{j}. Then since \frac{∂r}{∂s} = (1 − s)^{−2},

ω′ ≡ (Ψ^{−1})^{∗} (e^{−h}ν) =e^{− \frac{c}{2} (1−s)^{−2}}(1 − s)^{−2}u_{J}((1 − s)^{−1}, ȳ)ds ∧ dȳ^{J}
+ e^{− \frac{c}{2} (1−s)^{−2}}u_{I}((1 − s)^{−1}, ȳ)dȳ^{I}

is in Ω(M_{1}).

Denote as \frac{∂}{∂x^{j}} : Ω^{p}((0, ∞) × U_{j}) → Ω^{p}((0, ∞) × U_{j}) the operator defined by \frac{∂}{∂x^{j}} ω = ∑ \frac{∂ω_{I}}{∂x^{j}} dx^{I} if ω = ∑ ω_{I}dx^{I}. That is, \frac{∂}{∂x^{j}} differentiates the coefficients of ω with respect to x^{j}. If α = (α_{1}, …, α_{n}) ∈ Z_{+}^{n} is a multiindex, define \frac{∂}{∂x}^{α} = \frac{∂}{∂x^{1}} ^{α_{1}} … \frac{∂}{∂x^{n}} ^{α_{n}}.

Clearly, if

(2)  |r^{k} (∂/∂x)^{α} ν(r, x̄)| → 0 as r → ∞

for every k ≥ 0 and every multiindex α, then it follows that

(3)  | (∂/∂y)^{α} ω′(s, ȳ)| → 0 as s → 1^{−}

for every α, where (y^{1}, y^{2}, …, y^{n}) = (y^{1}, ȳ) = (s, ȳ) are coordinates on (0, 1) × U_{j}.

But (2) is exactly the conclusion of corollary 17, and (3) is exactly what is needed to prove property EZ on the end E. It follows from theorem 13 that j : H_{μ} → H_{c}(M) is surjective.

(Remark: We have proven somewhat more rapid decay in (3) than the minimum needed to prove property EZ. However, the degree to which the measure dμ can be changed to a measure dμ′ with slower growth rate e^{2h′} depends on both the growth rate e^{2h′} itself and the corresponding potential V = |dh′|^{2} + A_{h′} which plays the essential role in section 5.)

Proof that Π is surjective. Let {ω^{1}, …, ω^{N}} be an orthonormal basis for ker △_{μ}. Note N < ∞ since △_{μ} has compact resolvent. For each ω^{i}, we will find ν ∈ Ω_{c}(M) such that dν = 0 and Π(ν) = ω^{i}.

Choose ϵ = ϵ(N) > 0 so that any N × N real matrix (a_{ij}) with |a_{ij} − 1| < ϵ for all i and |a_{ij}| < ϵ for i ≠ j is invertible. For instance, ϵ = N^{−1} suffices.

In what follows we will make a finite number of choices for R, using the finite– dimensionality of ker △_{μ} and the finite list of ends E_{i}, and we assume that the maximum of these R is used as needed.

For any R ≥ 0, define Ψ_{R} analogously to Ψ on a particular end E: Ψ_{R}^{−1} is the identity on {s ≤ R} = M_{R}, Ψ_{R}^{−1}(s, x̄) = (R + (1 − (s − R))^{−1}, x̄) for R + 1/2 ≤ s < R + 1, and smoothly and monotonically interpolate Ψ^{−1} for R ≤ s ≤ R + 1/2. Thus Ψ_{R} is a diffeomorphism and Ψ_{R}(M) = M_{R+1}. We may assume |dΨ_{R}^{−1}| ≤ 4 on R ≤ s ≤ R + 1/2.

Next, we use property EZ: define j_{R} on forms φ ∈ C_{μ}^{∞} as the extension by zero of (Ψ_{R}^{−1})^{∗}φ. Let \overline{ω}^{i} = j_{R}ω^{i}. Each \overline{ω}^{i} ∈ Ω_{c}(M) and d\overline{ω}^{i} = (Ψ_{R}^{−1})^{∗}dω^{i} = 0, so [\overline{ω}^{i}] ∈ H_{c}(M).

Clearly | ⟨\overline{ω}^{i}, ω^{j}⟩_{μ} | ≤ ∥\overline{ω}^{i} − ω^{i}∥_{μ} if i ≠ j and | ⟨\overline{ω}^{i}, ω^{i}⟩_{μ} − 1| ≤ ∥\overline{ω}^{i} − ω^{i}∥_{μ}. We will show that we can choose R ≥ 0 so that ∥\overline{ω}^{i} − ω^{i}∥_{μ} < ϵ. It will follow that A = (⟨\overline{ω}^{i}, ω^{j}⟩_{μ}) is invertible.

Note \overline{ω}^{i}|_{M_{R}} = ω^{i}|_{M_{R}}. Thus

∥\overline{ω}^{i} − ω^{i}∥_{μ}^{2} = ∫_{R<s} |\overline{ω}^{i} − ω^{i}|^{2}dμ ≤ 2 ∫_{R<s<R+1} |\overline{ω}^{i}|^{2}dμ + 2 ∫_{R<s} |ω^{i}|^{2}dμ.

Now,

∫_{R<s<R+1/2} |\overline{ω}^{i}|^{2}dμ ≤ ∫_{R<s<R+1/2} |\frac{∂r}{∂s}|^{2} |ω^{i}(r(s), ȳ)|^{2}e^{cs^{2}}dsdȳ
≤ 4^{2} ∫_{R<r} |ω^{i}(r, x̄)|^{2}e^{cr^{2}}drdx̄

so the essential estimate is:

∫_{R+1/2<s<R+1} |\overline{ω}^{i}|^{2}dμ = ∫_{R+1/2<s<R+1} (1 − (s − R))^{−2} |ω^{i}(r(s), x̄)|^{2}e^{cs^{2}}dsdx̄
= ∫_{R+2<r<∞} |ω^{i}(r, x̄)|^{2} exp (c(R + 1 − (r − R)^{−1})^{2}) drdx̄
≤ ∫_{R+2<r<∞} |ω^{i}(r, x̄)|^{2} exp (c(R + 1)^{2}) drdx̄
≤ ∫_{R+1<r} |ω^{i}(r, x̄)|^{2}e^{cr^{2}}drdx̄.

It follows that

∥\overline{ω}^{i} − ω^{i}∥_{μ}^{2} ≤ 36 ∫_{R<r} |ω^{i}|^{2}dμ.

Choose R ≥ 0 so that ∥\overline{ω}^{i} − ω^{i}∥_{μ} < ϵ = ϵ(N) for every i = 1, …, N. Then A = (⟨\overline{ω}^{i}, ω^{j}⟩_{μ}) is invertible.

But

Π(\overline{ω}^{i}) = ∑_{j} ⟨\overline{ω}^{i}, ω^{j}⟩_{μ} ω^{j} = A_{ij}ω^{j},

so if ν = A_{ij}^{−1}\overline{ω}^{j} then

Π(ν) = A_{ij}^{−1}Π(\overline{ω}^{j}) = A_{ij}^{−1}A_{jk}ω^{k} = ω^{i}.

We have shown that Π is surjective, as in the method described in section 3. □

5. Schwartz Forms on a Cylindrical End

Let E = (0, ∞) × Q, where Q is compact without boundary. Give E a product metric, with (0, ∞) given the Euclidean metric. (Then E is not complete, but we will compute here as though it is, since for our purposes E is an end of a complete manifold.) Let Q = ⋃_{j} U_{j} be a finite cover of Q by coordinate neighborhoods. Let (x^{1}, x^{2}, …, x^{n}) = (x^{1}, x̄) be coordinates on (0, ∞) × U_{j}.

Let h = \frac{c}{2}(x^{1})^{2} for c > 0.

We define d_{h} = e^{h}de^{−h}, d_{h}^{∗} = e^{−h}de^{h}, and

△_{h} = d_{h}d_{h}^{∗} + d_{h}^{∗}d_{h} = △ + |dh|^{2} + A_{h},

all acting in L^{2}Ω (with measure dx = \sqrt{det g_{ij}}dx^{i}dx^{j} as usual). Then △_{h} on L^{2}Ω and △_{μ} on L_{μ}^{2}Ω are unitarily–equivalent ([Bi]).

Denote C_{h}^{∞} = ⋂_{k≥1} D_{(△_{h})^{k}}. Clearly, ω ∈ C_{h}^{∞} iff e^{−h}ω ∈ C_{μ}^{∞}. We will show that forms in C_{h}^{∞} decay as fast as Schwartz functions on a cylindrical end E as x^{1} → ∞.

Define ∂_{j} : Ω^{p}((0, ∞) × U_{j}) → Ω^{p}((0, ∞) × U_{j}) by ∂_{j}ω = ∑ \frac{∂ω_{I}}{∂x^{j}} dx^{I} if ω = ∑ ω_{I}dx^{I}. That is, ∂_{j} differentiates the coefficients of ω with respect to x^{j}. If α = (α_{1}, …, α_{n}) ∈ Z_{+}^{n} is a multiindex, let ∂^{α} = ∂_{1}^{α_{1}} … ∂_{n}^{α_{n}}.

We prove the following result on forms in C_{h}^{∞}:

16. Theorem. If ω ∈ C_{h}^{∞} is supported in E, then for any k ∈ N and any multiindex α,

(4)  ∥(x^{1})^{k}∂^{α}ω∥_{L^{2}Ω} < ∞.

Before proving (4), we note that it implies a pointwise result as well. That is, Sobolev inequalites (or even explicit Schwartz space calculations as in [RS], Appendix to V.3) imply:

17. Corollary. If ω ∈ C_{h}^{∞} is supported in E, then for any k ∈ N, any multiindex α, and any x̄_{0} ∈ U_{j},

(5)  |(x^{1})^{k}∂^{α}ω|(x^{1}, x̄_{0}) → 0 as x^{1} → ∞.

18. Remark. Properties (4) and (5) do not depend on the choice of local coordinates x̄ on Q, since all transition functions and their derivatives are bounded on compact Q.

Also, since Q is compact, by using a partition of unity on Q it is enough to prove theorem 16 only for forms which have support completely inside one of the coordinate neighborhoods (0, ∞)×U_{j}. Moreover, since inequality (4) is trivially true for smooth forms with compact support, we can assume (without loss of generality) that ω ≡ 0 for x^{1} ≤ 1.

The proof of theorem 16 will proceed in several steps, starting with following lemmata.

19. Lemma. Weitzenbock’s formula for △_{h}. Suppose ω is a smooth form on E, then

(6)  △_{h}ω = (∑_{i=1}^{n} ∇_{i}^{∗}∇_{i} + R(x̄) + c^{2}(x^{1})^{2} ± c) ω,

where R(x̄) : ∧^{p}T ∗ ; x̄E → ∧^{p}T ∗ ; x̄E is a pointwise curvature endomorphism, and ∇_{i} = ∇_{∂/∂x^{i}}.

Proof. See [CFKS], Chapter 12.4. □

20. Lemma. Gärding’s inequality. Let ω be a smooth form, suppose supp ω ⊂ (1, ∞) × U_{j} and suppose ω, △_{h}ω ∈ L^{2}Ω((1, ∞) × Q). Then

(7)  ∑_{i=1}^{n} ∥∇_{i}ω∥^{2} + ∥x^{1}ω∥^{2} ≤ C (⟨△_{h}ω, ω⟩ + |ω∥^{2}) .

Proof. Let J(τ) : (0, ∞) → R_{+} be a smooth non–increasing function such that J(τ) ≡ 1 for τ ≤ 2 and J(τ) ≡ 0 for τ ≥ 3. We define a family of cut–off functions

J_{t} : (0, ∞) → R_{+}

14 ED BUELER AND IGOR PROKHORENKOV

by

J_{t}(x^{1}, x̄) = J (\frac{x^{1}}{t} ) .

We multiply both sides of (6) by J_{t}^{2}ω and integrate by parts:

(8) ⟨△_{h}ω, J_{t}^{2}ω⟩ = ∑_{i=1}^{n} ⟨∇_{i}ω, ∇_{i}(J_{t}^{2}ω)⟩ + ⟨R(x̄)ω, J_{t}^{2}ω⟩ + c^{2}∥x^{1}J_{t}ω∥^{2} ± c∥J_{t}ω∥^{2}.

For each i we have

⟨∇_{i}ω, ∇_{i}(J_{t}^{2}ω)⟩ = ∥J_{t}∇_{i}ω∥^{2} + 2⟨J_{t} \frac{∂J_{t}}{∂x^{i}} ∇_{i}ω, ω⟩.

Therefore, substituting back into (8) we get

(9) ∑_{i=1}^{n}∥J_{t}∇_{i}ω∥^{2} + c^{2}∥x^{1}J_{t}ω∥^{2}

= ⟨△_{h}ω, J_{t}^{2}ω⟩ − 2 ∑_{i=1}^{n} ⟨J_{t} \frac{∂J_{t}}{∂x^{i}} ∇_{i}ω, ω⟩ − ⟨R(x̄)ω, J_{t}^{2}ω⟩ ± c∥J_{t}ω∥^{2}.

Now we estimate:

2|⟨J_{t} \frac{∂J_{t}}{∂x^{i}} ∇_{i}ω, ω⟩| ≤ sup_{x̄} |\frac{∂J_{t}}{∂x^{i}} |^{2} ∥J_{t}∇_{i}ω∥^{2} + ∥ω∥^{2}

≤ \frac{C}{t^{2}} ∥J_{t}∇_{i}ω∥^{2} + C∥ω∥^{2}

and

|⟨R(x̄)ω, J_{t}^{2}ω⟩| ≤ sup_{x̄} |R(x̄)|∥J_{t}ω∥^{2} ≤ C∥ω∥^{2}.

For large t, the two inequalities above, together with (9), give:

(10) ∑_{i=1}^{n} ∥J_{t}∇_{i}ω∥^{2} + c^{2}∥x^{1}J_{t}ω∥^{2} ≤ C (⟨△_{h}ω, J_{t}^{2}ω⟩ + ∥J_{t}ω∥^{2})
≤ C (⟨△_{h}ω, ω⟩ + ∥ω∥^{2}) .

Gärding’s inequality now follows if we take t → ∞ in the inequality above. □

21. Lemma. Let ω be a smooth form. Suppose supp ω ⊂ (1, ∞) × U_{j} and suppose ω, △_{h}ω ∈ L^{2}Ω((1, ∞) × Q). Then

(11)  ∑_{j=1}^{n} ∥J_{t}∂_{j}ω∥^{2} + ∥J_{t}x^{1}ω∥^{2} ≤ C (| ⟨J_{t}^{2}△_{h}ω, ω⟩ | + ∥J_{t}ω∥^{2}) ,

and

(12)  ∑_{j=1}^{n} ∥∂_{j}ω∥^{2} + ∥x^{1}ω∥^{2} ≤ C (⟨△_{h}ω, ω⟩ + ∥ω∥^{2}) .

Proof. Recall that

(13)  ∇_{i}(ω_{I}dx^{I}) = (∇_{i}ω_{I})dx^{I} + ω_{I}∇_{i}(dx^{I}) = (∂_{i}ω_{I})dx^{I} + ω_{I}Γ_{i}(x̄)(dx^{I}),

where Γ_{i}(x̄) : ∧T ∗ ; x̄ → ∧T ∗ ; x̄ is an endomorphism. The matrix of Γ_{i}(x̄) can be expressed in terms of the Christoffel symbols Γ_{ij}^{k}(x̄).

It follows from (13) that

(14)  ∥J_{t}∇_{j}ω∥^{2} = ∥J_{t}∂_{j}ω∥^{2} + 2 ⟨J_{t}Γ_{j}(x̄)ω, J_{t}∂_{j}ω⟩ + ∥J_{t}Γ_{j}(x̄)ω∥^{2}.

For some choice of small ϵ > 0, we estimate:

2| ⟨J_{t}Γ_{j}(x̄)ω, J_{t}∂_{j}ω⟩ | ≤ \frac{1}{ϵ}∥J_{t}Γ_{j}ω∥^{2} + ϵ∥J_{t}∂_{j}ω∥^{2} ≤ \frac{C}{ϵ} ∥J_{t}ω∥^{2} + ϵ∥J_{t}∂_{j}ω∥^{2}.

Therefore, from (14)

(15)  ∥J_{t}∂_{j}ω∥^{2} ≤ C (∥J_{t}∇_{j}ω∥^{2} + ∥J_{t}ω∥^{2}) .

Inequality (11) follows from (10) and (15). Inequality (12) follows from (7) and (15) as t → ∞. □

Proof of theorem 16. Induction argument. For ω ∈ C_{h}^{∞}, supp ω ⊂ (0, ∞) × U_{j}, inequality (4) follows from

(16)  ∥ω∥_{l}^{2} := ∑ ; \begin{matrix}k, α \\ k + |α| ≤ l \end{matrix} ∥(x^{1})^{k}∂^{α}ω∥^{2} ≤ C(l) (⟨△_{h}^{l}ω, ω⟩ + ∥ω∥^{2})

where 0 ≤ l < ∞.

Remark. Operator △_{h}^{l} is a PDO of order 2l with smooth coefficients which grow at most as (x_{1})^{2l}. Therefore, for all ω ∈ C_{h}^{∞} which satisfy ∥ω∥ ≤ ∞, we have

(17)  ⟨△_{h}^{l}ω, ω⟩ ≤ C_{l}∥ω∥_{l}^{2}.

Moreover, for all l_{0} ≤ l

⟨△_{h}^{l_{0}}ω, ω⟩ ≤ C (⟨△_{h}^{l}ω, ω⟩ + | ω∥^{2}) .

We will prove (16) by induction in l. Lemma 21 proves (16) for l = 1 and all ω ∈ C_{h}^{∞} with supp ω ⊂ (0, ∞) × U_{j}.

Suppose (16) is true for all l ≤ l_{0} and all ω as above. Let k + |α| = l_{0}. We substitute (x^{1})^{k}∂^{α}ω instead of ω into (11) to get

(18)  ∑_{j=1}^{n}∥J_{t}∂_{j}(x^{1})^{k}∂^{α}ω∥^{2} + ∥J_{t}(x^{1})^{k+1}∂^{α}ω∥^{2}
≤ C (|⟨J_{t}^{2}△_{h} ((x^{1})^{k}∂^{α}ω) , (x^{1})^{k}∂^{α}ω⟩| + ∥J_{t}(x^{1})^{k}∂^{α}ω∥^{2}) .

Define A = ⟨J_{t}^{2}△_{h} ((x^{1})^{k}∂^{α}ω) , (x^{1})^{k}∂^{α}ω⟩. We write

(19)  △_{h} ((x^{1})^{k}∂^{α}ω) = (x^{1})^{k}∂^{α}△_{h}ω + [△_{h}, (x^{1})^{k}∂^{α}] ω,

where [·, ·] denotes the commutator of two differential operators. Then from (6),

(20) [△_{h}, (x^{1})^{k}∂^{α}] ω = [∑_{i=1}^{n} ∇_{i}^{∗}∇_{i} + R(x̄), (x^{1})^{k}∂^{α}] ω + c^{2} [(x^{1})^{2}, (x^{1})^{k}∂^{α}] ω
= P_{1}ω + P_{2}ω.

Proposition.

a) P_{1} is a PDO of order at most |α| + 1 with coefficients which grow at most as (x^{1})^{k}.

b) P_{2} is a PDO of order at most |α| with coefficients which grow at most as (x^{1})^{k+1}.

Proof of Proposition. This follows from the formula (13) for ∇_{i} and from two simple observations.

First,

[(x^{1})^{k}, ∂_{j}] = \{ \begin{matrix}0, & j ≠ 1 \\ k(x^{1})^{k−1}, & j = 1. \end{matrix}

Second, the commutator of a smooth function f(x̄) with ∂^{α} is a PDO of order |α| − 1 with smooth (bounded) coefficients which depend only on x̄. Q.E.D.

With the help of (19) and (20) we rewrite the quantity A as

(21) A = ⟨J_{t}^{2}(x^{1})^{k}∂^{α}△_{h}ω, (x^{1})^{k}∂^{α}ω⟩+⟨J_{t}^{2}P_{1}ω, (x^{1})^{k}∂^{α}ω⟩+⟨J_{t}^{2}P_{2}ω, (x^{1})^{k}∂^{α}ω⟩ ,

and we estimate:

(22) |A| ≤ ∥J_{t}(x^{1})^{k}∂^{α}△_{h}ω∥ · ∥J_{t}(x^{1})^{k}∂^{α}ω∥

+ ϵ∥J_{t}P_{1}ω∥^{2} + \frac{1}{ϵ}∥J_{t}(x^{1})^{k}∂^{α}ω∥^{2} + ϵ∥J_{t}P_{2}ω∥^{2} + \frac{1}{ϵ}∥J_{t}(x^{1})^{k}∂^{α}ω∥^{2},

HODGE THEORY AND COHOMOLOGY WITH COMPACT SUPPORTS 17

where ϵ > 0 is to be chosen later. Now we estimate the right–hand side of (22). For △_{h}ω ∈ C_{h}^{∞}, by induction and by the remark above we have

∥J_{t}(x^{1})^{k}∂^{α}△_{h}ω∥ · ∥J_{t}(x^{1})^{k}∂^{α}ω∥ ≤ C (⟨(△_{h})^{l_{0}+1}ω, ω⟩ + ∥ω∥^{2}) .

Moreover, from the proposition,

∥J_{t}P_{1}ω∥^{2} + ∥J_{t}P_{2}ω∥^{2} ≤ C ∑_{m+|β|≤l_{0}+1} ∥J_{t}(x^{1})^{m}∂^{β}ω∥^{2}.

Therefore,

|A| ≤ C( ⟨(△_{h})^{l_{0}+1}ω, ω⟩ + ∥ω∥^{2} + ϵ · ∑_{m+|β|≤l_{0}+1} ∥J_{t}(x^{1})^{m}∂^{β}ω∥^{2} + ∥ω∥_{l_{0}}^{2}).

The inequality (18) can be rewritten as

∑_{j=1}^{n}∥J_{t}(x^{1})^{k}∂_{j}∂^{α}ω∥^{2} + ∥J_{t}(x^{1})^{k+1}∂^{α}ω∥^{2} (23)

≤ C( ⟨(△_{h})^{l_{0}+1}ω, ω⟩ + ∥ω∥^{2} + ∥ω∥_{l_{0}}^{2}) + Cϵ · ∑_{m+|β|≤l_{0}+1} ∥J_{t}(x^{1})^{m}∂^{β}ω∥^{2}.

We can sum up inequalities (23) over all k + |α| = l_{0} and use the equality

∑_{k+α≤l_{0}} ∑_{j=1}^{n} ∥J_{t}(x^{1})^{k}∂_{j}∂^{α}ω∥^{2} + ∥J_{t}(x^{1})^{k+1}∂^{α}ω∥^{2} = ∑_{m+|β|≤l_{0}+1} ∥J_{t}(x^{1})^{m}∂^{β}ω∥^{2}

to get (24) from (23):

(24) ∑_{m+|β|≤l_{0}+1} ∥J_{t}(x^{1})^{m}∂^{β}ω∥^{2} ≤ C(l_{0} + 1)( ⟨(△_{h})^{l_{0}+1}ω, ω⟩ + ∥ω∥^{2} + ∥ω∥_{l_{0}}^{2})

+ ϵ C(l_{0} + 1)( ∑_{m+|β|≤l_{0}+1} ∥J_{t}(x^{1})^{m}∂^{β}ω∥^{2}).

Finally, we choose ϵ small enough to move the right–hand ∑_{m+|β|≤l_{0}+1} over to the left.

Then (16) follows from (24) for l = l_{0} + 1 if we take t → ∞. □

6. Application: Künneth Formula

22. Proposition. Let M, N be orientable topologically tame manifolds. Then

H_{c}(M × N) ≅ H_{c}(M) ⊗ H_{c}(N).

Proof. Choose metrics g_{M}, g_{N} so that M, N have cylindrical ends as in theorem 2, and define dμ_{M}, dμ_{N} as described there, choosing c > 0. Apply theorem 15 to M, N, and note that the same methods used to prove theorem 15 also show that H_{c}(M × N) ≅ H_{μ_{M} ×μ_{N}}, where M × N has been given the product metric and the product measure.

Now, H_{μ_{M} ×μ_{N}} has the expected combinatorics, coming purely from analysis on Hilbert tensor products—more precisely Fredholm complexes. See [BL]. Specifically,

H_{μ_{M} ×μ_{N}} ≅ H_{μ_{M}} ⊗ H_{μ_{N}} ,

which proves our theorem. □

The above theorem applies only to what [BT] call manifolds with “finite good cover”, but it is nonetheless a Hodge–theoretic proof of a Künneth formula for certain noncompact manifolds. See [GH], chapter 0, for proof of the Künneth formula on compact manifolds, using the same Hodge–theoretic technique of proof. The Künneth formula in fact holds for any pair of manifolds M, F for which the cohomology of F is finite–dimensional.

References

[An] Michael T. Anderson, L^{2} harmonic forms on complete Riemannian manifolds, in “Geometry and Analysis on Manifolds: Proceedings Katata-Kyoto 1987,” (Toshikazu Sunada, ed.) Springer, Berlin, 1988.

[At] M. F. Atiyah, Elliptic operators, discrete groups and von Neumann algebras, Astérisque 32–33 (1976), 43–72.

[APS] M. F. Atiyah, V. K. Patodi, & I. M. Singer, Spectral asymmetry and Riemannian geometry. I, Math. Proc. Camb. Phil. Soc. 77 (1975), 43–69.

[Bi] Jean-Michel Bismut, The Witten complex and the degenerate Morse inequalities, J. Differential Geom. 23 (1986), 207–240.

[BT] Raoul Bott & Loring W. Tu, Differential Forms in Algebraic Topology, Springer, New York, 1982.

[BL] J. Brüning & M. Lesch, Hilbert complexes, J. Funct. Anal. 108 (1992), 88–132.

[Bu] Edward L. Bueler, The heat kernel weighted Hodge Laplacian on noncompact manifolds, Trans. Amer. Math. Soc. to appear (1998).

[C] Paul R. Chernoff, Essential self–adjointness of powers of generators of hyperbolic equations, J. Funct. Anal. 12 (1973), 401–414.

[CFKS] Hans L. Cycon, Werner Kirsch, Richard G. Froese, & Barry Simon, Schrödinger Operators with Application to Quantum Mechanics and Global Geometry, Springer, Berlin, 1987.

[G] Matthew P. Gaffney, Hilbert space methods in the theory of harmonic integrals, Trans. Amer. Math. Soc. 78 (1955), 426–444.

[GH] Phillip Griffiths & Joseph Harris, Principles of Algebraic Geometry, Wiley, New York, 1978.

[L] John Lott, The zero–in–the–spectrum question, Enseign. Math. (2) 42 (1996), 341–376.

[M] Richard B. Melrose, The Atiyah–Patodi–Singer Index Theorem, A. K. Peters, Wellesly, Massachusetts, 1993.

[P] Igor Prokhorenkov, Morse–Bott functions and the Witten Laplacian, Preprint (1997).

[RS] Michael Reed & Barry Simon, Methods of Modern Mathematical Physics, Volume I: Functional Analysis, Revised Edition, Academic Press, New York, 1980.

[Wa] Frank W. Warner, Foundations of Differentiable Manifolds and Lie Groups, Springer, New York, 1983.

[Wi] Edward Witten, Supersymmetry and Morse theory, J. Differential Geom. 17 (1982), 661–692.

Ed Bueler — Mathematical Sciences Research Institute, Berkeley

   E-mail address: ebueler@msri.org

Igor Prokhorenkov — Current address: Kosmonavtov 31/3–94, Lipetsk, 398032, Russia. After 9/1/98: Department of Mathematics, Texas Christian University, Fort Worth, TX.

   E-mail address: prokhor@mccme.ru