Witten deformation for noncompact manifolds with bounded geometry

Motivated by the Landau-Ginzburg model, we study the Witten deformation on a noncompact manifold with bounded geometry, together with some tameness condition on the growth of the Morse function $f$ near infinity. We prove that the cohomology of the Witten deformation $d_{Tf}$ acting on the complex of smooth $L^2$ forms is isomorphic to the cohomology of Thom-Smale complex of $f$ as well as the relative cohomology of a certain pair $(M, U)$ for sufficiently large $T$. We establish an Agmon estimate for eigenforms of the Witten Laplacian which plays an essential role in identifying these cohomologies via Witten's instanton complex, defined in terms of eigenspaces of the Witten Laplacian for small eigenvalues. As an application we obtain the strong Morse inequalities in this setting.


Overview
In the extremely influential paper [19], Witten introduced a deformation of the de Rham complex by considering the new differential d f = d + df, where d is the usual exterior derivative on forms, and f is a Morse function. Setting Witten observed that when T > 0 is large enough, the eigenfunctions of the small eigenvalues for the corresponding deformed Hodge-Laplacian, the so called Witten Laplacian, concentrate at the critical points of f . As a result, Witten deformation builds a direct bridge between the Betti numbers and the Morse indices of the critical points of f . Witten deformation on closed manifolds has produced a whole range of beautiful applications, from Demailly's holomorphic Morse inequalities [14], to the proof of Ray-Singer conjecture and its generalization by Bismut-Zhang [2], to the instigation of the development of Floer homology theory.
Although the Witten deformation on noncompact manifolds are much less studied and understood, there are previous interesting work in the direction. In [4] the cohomology of an affine algebraic variety is related to that of the Witten complex of C m , see also [7] for further development.
This paper is motivated by the study of Landau-Ginzburg models (c.f. [12]), which, according to Witten [20], are simply different phase of Calabi-Yau manifolds, and hence equivalent to Calabi-Yau manifolds. Suppose there is a non-trivial holomorphic function W (the superpotential) on a noncompact Kahler manifold M n (n = dim C M ), then one considers the Wittendeformation of ∂ operator:∂ as its cohomology describes the quantum ground states of the Landau-Ginzburg model (M, W ). If W is also a Morse function with k critical points, then complex Morse theoretic consideration leads to the expectation that For the mathematical study of LG models and their significant applications we point out the important work [9]. In this paper, we consider the more general case for Riemannian manifolds: we explore the relations between the Thom-Smale complex for a Morse function f on a noncompact manifold M and the deformed de Rham complex with respect to f . The first difficulty one encounters here is the presence of continuous spectrum on a noncompact manifolds and for that one has to impose certain tameness conditions. This consists of the bounded geometry requirement for the manifold as well as growth conditions for the function. The notion of strong tameness is introduced in [5] in the Kähler setting which guarantees the discreteness of the spectrum for the Witten Laplacian. Here we introduce a slightly weaker notion which allows continuous spectrum but only outside a large interval starting from 0.
It is important to note that, and this is another new phenomenon in the noncompact case, the Thom-Smale complex may not be a complex in general. Namely, the square of its boundary operator need not be zero, since M is noncompact. However we prove that with the tameness condition, it is.
The crucial technical part of our work is the Agmon estimate for eigenforms of the Witten Laplacian which is essential in extending the usual analysis from compact setting to the noncompact case. The Agmon estimate was discovered by S. Agmon in his study of N -body Schrödinger operators in the Euclidean setting and has found many important applications. The exponential decay of the eigenfunction is expressed in terms of the so-called Agmon distance, Cf. [1]. We make essential use of this Agmon estimate to carry out the isomorphism between the Witten instanton complex defined in terms of eigenspaces corresponding to the small eigenvalues with the Thom-Smale complex defined in terms of the critical point data of the function. We remark that the Agmon estimate near the critical points also plays important role in the compact case , see, e.g. [2]. The novelty here is that we make essential use of the exponential decay at spatial infinity provided by the Agmon estimate.
As an application of our results on noncompact manifolds, we deduce corresponding results for manifolds with boundaries which generalize recent work of [16], [?].
Finally we would also like to point out the preprint [6] which has provided further motivation and inspiration for us.
In the rest of the introduction we give precise statements of our main results after setting up our notations. In subsequent work we will develop the local index theory and the Ray-Singer torsion for the Witten deformation in the noncompact setting.
Acknowledgment: We thank Shu Shen for interesting discussions and for providing us with an example of Thom-Smale complex not being a complex.

Notations and basic setup
Let (M, g) be a noncompact connected complete Riemannian manifold with metric g. (M, g) is said to have bounded geometry, if the following conditions hold: 1. the injectivity radius r 0 of M is positive.
2. |∇ m R| ≤ C m , where ∇ m R is the m-th covariant derivative of the curvature tensor and C m is a constant only depending on m.
On such a manifold, the Sobolev constant is uniformly bounded, see e.g. [13]. Now let f : M → R be a smooth function. In [5], the notion of strong tameness for the triple (M, g, f ) is introduced. In this paper we only need the following weaker condition.
Definition 1.3. The triple (M, g, f ) is said to be well tame, if (M, g) has bounded geometry and As usual, the metric g induced a canonical metric (still denote it by g) on Λ * (M ), which then defines an inner product (·, ·) L 2 on Ω * c (M ): Let L 2 Λ * (M ) be the completion of Ω * c (M ) with respect to · L 2 , and for simplicity, we denote L 2 (M ) := L 2 Λ 0 (M ).
For any 2 , and we denote the Friedrichs extension of ∆ H,T f by T f . As we will see (Theorem 2.1), if (M, g, f ) is well tame, then ∆ H,T f is essentially self-adjoint (and hence T f is the unique self-adjoint extension). In Section 8, we will prove the Hodge-Kodaira decomposition when (M, g, f ) is well tame and T large enough, Let H * (2) (M, d T f ) denote the cohomology of this complex. In Section 8, we will show that H * (2) (M, d T f ) ∼ = ker T f , provided (M, g, f ) is well tame and T is large enough.
Finally we note the following well known (Cf. [19,21]) Proposition 1.4. The Hodge Laplacian ∆ H,T f has the following local expression: Here {e i } is a local frame on T M and {e i } is the dual frame on T * M.

Main results
In this subsection, we assume that (M, g) has bounded geometry, f is a Morse function with finite many critical points. Clearly this will be the case if (M, g, f ) is well tame and f is Morse.
As we mentioned the main technical result here is the Agmon estimate for the eigenforms of the Witten Laplacian. Theorem 1.1. Let (M, g, f ) be well tame, and ω ∈ Dom( T f ) be an eigenform of T f whose eigenvalue is uniformly bounded in T . Then for any a ∈ (0, 1) (provided T is sufficiently large and C is a constant depending on the dimension n, the function f , the curvature bound, and a; for the precise choice of T, C see the end of Section 3). Here the definition of the Agmon distance ρ T (p) will be given in Section 3.
The proof of the Agmon estimate, given in Section 7, is to carry out the idea of [1] in this more general setting. Set Let m i be the number of critical points of f with Morse index i. Then the strong Morse inequalities hold.
is well tame, then we have the following strong Morse inequality provided T is large enough. And the equality holds for k = n.
In general, b i (T ) may be very sensitive to T . However we have the following result regarding the indepedence of b i (T ) in T . Assume that the Morse function f satisfies the Smale transversality condition. Let (C * (W u ),∂ ′ ) be the Thom-Smale complex given by f . It is important to note that in general, since M is noncompact, it could happen that (∂ ′ ) 2 = 0. Also let c > 0 be big enough, U c = {p ∈ M : f (p) < −c} and (Ω * (M, U c ), d) be the relative de Rham complex. As an another application of Theorem 1.3, we study the Morse cohomology for compact manifolds with boundary.
Let M be a compact, oriented manifold of dimension n with boundary ∂M . Let N i (i ∈ Λ) be the connected components of ∂M . We fix a collar neighborhood (0, 1] × N i ⊂ M , and let r be the standard coordinate on the (0, 1] factor. is isomorphic to the relative de Rham cohomology H * (M, ∂M ).
, then we have the following Morse inequalities:

Notations and Organization
In this article, we will generally use φ, ψ to denote differential forms, f Morse function, u, v functions, ν, ω eigenforms, x, y, z critical points of f , p, q general points, and p 0 a fixed point. This paper is organized as follows. In Section 2, we discuss the spectral theory for the Witten Laplacian in our setting. We then proceed to establish the exponential decay estimate for eigenforms of the Witten Laplacian in Section 3. Assuming two technical results whose proofs are deferred to Section 7 (7.3) and using a lemma proved in Section 5 about the Agmon distance we prove Theorem 1.1, the Agmon estimate.
In Section 4 we present the proof of the Strong Morse Inequalities, Theorem 1.2, after introducing Witten instanton complex (F , d T f ). Section 5 concerns the Thom-Smale theory in our setting. More specifically, we define the Thom-Smale complex C * ((W u ) ′ ,∂ ′ ). Then we define a morphism between the Witten instanton complex and the Thom-Smale complex, J : (F . We prove that J is well defined by using the Agamon estimate, deferring the proof that J is a chain map to Subsection 7.3.
In Section 6 we give an application of our results, namely Theorem 1.5. Section 7 collects the proofs of several technical results. In the first two subsections we prove the lemmas used in the proof of Agmon estimate. In the Subsection 7.3, we prove that our Thom-Smale complex is indeed a complex, i.e.,∂ ′2 = 0. The rest of the proof for Theorem 1.3 is in Subsection 7.4 and Subsection 7.5. Finally in Section 8, which is an appendix, we discuss the Kodaira decomposition in a more general setting.

The Spectrum Of Witten Laplacian
In this section we study the spectral theory of the Witten Laplacian on noncompact manifolds. In particular we establish the Kodaira decomposition and the Hodge theorem for the Witten Laplacian under our tameness condition.

Essential self-adjointness of
Proof. Since lim sup p→∞ |∇f | 2 (p) < ∞, f is bounded from below by Proposition 1.4. The rest of the proof is essentially the same as in Section 4 of [3]; see also the proof of Theorem 1.17 in [8].

On the spectrum of T f
From now on we will assume that (M, g, f ) is well tame. Then it follows that there exists a compact subset K, which can be taken to be a compact submanifold with boundaries that contains the closure of a ball of sufficiently large radius of M (we will make a more specific choice of K later in section 5), (2.1) First, we establish the following basic lemma.
Here C R is a constant depending only on the sectional curvature bounds of g.
Proof. It suffices to show the inequality for a compactly supported smooth form. By Proposition 1.4, together with the Bochner-Weitzenböck formula, we have Thus, for any b ∈ (0, 1), there exists Remark 2.2. When (M, g, f ) is strongly tame, we can take T 1 = 0, but K may depend on T.
Letg T := b 2 T 2 |∇f | 2 g be a new metric on M (with discrete conical singularities).Fix p 0 ∈ K, let ρ T (p) be the distance between p and p 0 induced byg T . Then we have |∇ρ T | 2 = b 2 T 2 |∇f | 2 a.e., where the gradient ∇ is induced by g.
consists of a finite number of eigenvalues of finite multiplicity.
Proof. Let P : Combining with (2.2), we have Furthermore, the Hodge Theorem holds:

Exponential decay of eigenfunction
In this section, we assume that (M, g, f ) is well tame, and T ≥ T 2 , where T 2 is described in Lemma 2.1. If (M, g, f ) is strongly tame, then we can just take T 2 = 0. Recall thatg T := b 2 T 2 |∇f | 2 g, the Agmon metric on M. Let K be the compact set as in last section. In this and later sections we define the Agmon distance ρ T (p) be the distance between p and K induced byg T .
For simplicity, denote b 2 T 2 |∇f | 2 by λ T . We need the following two technical lemmas, whose proofs are postponed to Section 7.
Then there exists another compact subset Corollary 3.1. If w = cu for some c > 0 and T ≥ 2 √ 1+c bǫ f , then Proof. With this choice of T , λ T > 1 + c outside K. Now replacing λ T with λ T − c and w with 0 in Lemma 3.1, we get By refining the argument above, we have the following corollary which will be used in the proof of our Agmon estimate for eigenforms.
Proof. Following the proof of Lemma 3.1 given in Section 7.
We split the second integral on the right hand side into two; the one over L − K will be absorbed into the first term. The second term is (we omit the volume form here) Combining the above one arrives at and consequently Remark 3.2. It may seem that C L and C L depend on T as L = {p ∈ M : ρ T (p) < r 0 }. However, notice that when T becomes bigger, L gets smaller. Hence we can choose C L > max p∈L |∇f |(p), C L > max p∈L |∇ 2 f (p)|, s.t. they are independent of T.

Lemma 3.3 (De Giorgi-Nash-Moser Estimates).
For r > 0, let B r (p) be the geodesic ball around p with radius r (in the metric g). Let 0 ≤ u ∈ L 2 (M ), and ∆u ≤ cu on B 2r (p) in the weak sense for some constant c ≥ 0.
Then there exists constant C 2 > 0 depending only on the dimension n, the Sobolev constant, and c, such that With these preparation we are now ready to prove our first main estimate for the eigenforms of T f .
Then letting u = g(ω, ω) 1/2 , by a straightforward computation using the Bochner's formula (for forms) and the Kato's inequality, we have where |R| is the upper bound of curvature tensor. Hence by Corollary 3.2, Recall that the compact set K is chosen so that (2.1) is satisfied. Hence by Proposition 1.4, the conditions of Lemma 3.3 are satisfied for u on M − K. Also, the Agmon distance ρ T (p) is the distance between p and K induced byg T and L = {p ∈ M : ρ T (p) ≤ 2}. Suppose p ∈ M − L. Denote byB r (p) theg T -geodesic ball around p with radius r. Set l = sup q∈B 2 (p) |T ∇f |(q), and r = 1/(2l). Then one can easily verify that Choose q 0 ∈B 2 (p) so that |T ∇f |(q 0 ) ∈ (l/2, l]. By Lemma 3.1 and de Giorgi-Nash-Moser estimate Lemma 3.3, we have We will prove that in Lemma 5.6. Hence, for any small ǫ, provided T ≥ 2c f bǫ . It follows then that, Remark 3.4. The proof above gives the inequality for p ∈ M − L = {ρ T (p) > r 0 } for some constant r 0 independent of T , which is what we needed for later applications. For p ∈ L, using the same reasoning as in for all p ∈ L. Therefore via Moser iteration as in Lemma 3.3 and similar arguments as above, one can show that

Morse inequalities
In this and the next section, we assume that f is a Morse function on M , and T ≥ T 0 . In fact, we assume that in a neighborhood U x of critical points x of f , we have coordinate system z = (z 1 , ..., z n ), such that This is a generic condition. Without loss of generality we assume that U Here supp(φ) denotes the support of φ.
On the other hand, (F

Thom-Smale theory
In this section, we assume that K is a compact subset of M , ǫ > 0 is small enough (to be determined later), Moreover, we make a more judicious choice of K. Fix any p 0 ∈ M . Set whered T is the distance function induced byg T (note that the Agmon distance ρ T (x) is the same distance function but between x and a compact subset K). The second equality follows from the claim in the proof of Lemma 5.5. We choose K so that that where K • denotes the interior of K. For this purpose, we fix a positve smooth function F such that Then we have Proof. LetΦ t be the flow generated by −F ∇f . We show that for any p ∈ M , there exists a universal ǫ 0 > 0, s.t.Φ t (p) is well defined on (−ǫ 0 , ǫ 0 ). Hence, −F ∇f is complete.
Let L := {p ∈ M :d T (p, K) ≤ 1}. It suffices to show that for any Let x be a critical point of the Morse function f , W s (x) and W u (x) be the stable and unstable manifold of x with respect to flowΦ t defined in Lemma 5.2 (See Chapter 6 in [21]). We will further assume that f satisfies the Smale transversality condition, namely W s (x) and W u (y) intersect transversally. Then the Thom-Smale complex (C * (W u ), ∂) is defined by To define the boundary operator, let x and y be critical points of f , with Here the integer m(x, y) is the signed counts of the flow lines in W s (y) ∩ W u (x),.
Remark 5.3. With our nice choice of K and F , it is easy to see that for any x, y ∈ Crit(f ), W s (x) ∩ W u (y) ⊂ K • . Moreover, for any p ∈ W u (x) − K, the curve {Φ t (p) : t ≥ 0} ∩ K = ∅ and lim t→∞ |f |(Φ t (p)) ≈ lim t→∞ ρ T (Φ t (p)) = ∞. Moreover, sinced T (p, p 0 ) = Td 1 (p, p 0 ), we can actually choose K, such that it is independent of T. Thus, just like the compact case, by the transversality, m(x, y) is well defined.
We will prove in Section 7.3 that under our tameness condition,∂ 2 = 0. Thus, C * (W u ,∂) is a complex.
Recall that the Witten instanton complex F [0,1], * T f is the finite dimensional space generated by the eigenforms of T f with eigenvalue lying in [0, 1]. By the discussion in the previous subsection, the cohomology of the Witten instanton comple is H * (2) (M, d f ). To prove Theorem 1.3, we now consider the following chain map J : However there is a technical issue here we need to address. When W u (x) is compact, the integral W u (x) exp(T f )ω is clearly well defined, but W u (x) here may be noncompact. We will be content here only with the weldefinedness of the map and leave the proof that J is indeed a chain map to Section 7.3.
Let r > 0 small enough, B with center x and radius r with respect to metric g. As before, let Φ t be the flow generated by −F ∇f .
The well definedness of J is now reduced to the following two technical lemmas, as well as Theorem 1.1 and the well tameness of (M, g, f ).
Here C 7 is a constant independent of y.
Proof. Let e be a unit tangent vector of W u (x) at y, Extend e to a local unit vector field near y via parallel transport along radial geodesics. Denote Noting that from (5.1) By a classical result in ODE, we have Now our lemma follows from teh following, Lemma 5.5.
Let∇ T be the Levi-Civita connection induced byg T , theñ We now prove that γ is the shortest geodesic connecting p and γ(r) in (M,g T ), for all r > 0: Assume that σ is another normal geodesic connecting p and γ(r) in- Hence by a comparison theorem in ODE, we must have a(s) ≤ b(s). Assume that σ(r ′ ) = γ(r), then we can see r ′ = Length(σ), also we have a(r ′ ) ≤ b(r ′ ) = a(r). Since a is decreasing, we must have r ′ ≥ r.
By now, we can see thatΦ s (y), s ∈ [0, t] is one of shortest geodesic connecting y andΦ t (y).
We now note the following lemma which plays an important role in estimating the eigenforms previously.
Here we are gives a direct proof of the isomorphism of H * (2) (M, d T f ) and H * dR (M, U c ) under the assumption that f is proper. Proof. We may as well set K = f −1 [I, S]. Motivated by [7], consider .
By Theorem 1.1, and similar argument in Lemma 5.5, we can see that Hence L induces a homomorphism (still denote it by L) between H * (Ω • (2) (M ), d T f ) and H * (Cone • , d C ). The proof of the fact that L is a bijection is tedious, which will be given in Subsection 7.6. Then we can extend the metric g toM , s.t. near the infinity, the metric g onM is of product type, i.e. g ∂M + dr 2 . It's easy to see that (M , g) has bounded geometry. We have the following technical lemma Lemma 6.1. Given a transversal Morse function f , a partition of bound- We are able to extend f to a functionf onM , s.t.

The Agmon Estimate
In this section we carry out the main technical estimates of the paper.

Proof of Lemma 3.1
Proof. Our proof is adapted from Theorem 1.5 in [1].
As a consequence, We state two Lemmas that would be needed shortly, t. T f u ≤ w in weak sense (Here we assume u ≥ 0.). For r > 0 small enough, p / ∈ L, let B r (p) be the geodesic ball around p with radius r induced by g. Then there exist C 2 > 0, where C 2 is a constant that depends only on dimension n.
Proof. The proof is actually similar to the proof Lemma 3.3, but a little more complicated. See Theorem 4.1 in [10] for a reference.
By the same argument as the proof of Theorem 1.1, we have for some a ′′ ∈ (0, b). If φ ∈ L 2 (M ) is a weak solution of T f φ ≤ w, then |φ(p)| ≤ C exp(−a ′′ ρ T (p)).

On the Thom-Smale complex
First, let's recall the situation of the compact case. The following is a restatement of Proposition 6 in [15]. Let (N, g) be a compact Riemannian manifold, f be a Morse function. Assume that (N, g, f ) satisfies Thom-Smale transversality condition. Then, for any critical point x ∈ Crit(f ) with Morse index n f (x), any φ ∈ Ω n f (x)−1 (M ), one has the following so called Stokes Formula For our noncompact case with tame conditions and Thom-Smale transversality, we have similarly (M ), one has the following so called Stokes Formula Before giving the proof of this proposition, we first draw a couple of consequneces.

Now our Corollary follows from Proposition 7.4 trivially.
Hence, the map J introduced in Section 5 is a chain map. The proof of Proposition 7.4 follows from the following observations: The proof is essentially the same with Theorem 2.5 in [18]. Observation 7.6. LetB D (p 0 ) be the ball with radius D introduced in Section 5. Then for any y, z ∈ Crit(f ), W u (y) ∩ W s (z) ⊂B • D . Moreover, if p / ∈B D (p 0 ) lies in an unstable manifold, then the curve {Φ t (p) : t ≥ 0} ∩B D = ∅. HereB • D denotes the interior ofB D =B D (p 0 ), Φ t is the flow generated by −∇f.
Proof. It follows easily from the fact that for any p ∈ M ,d T (p, Φ t (p)) = |f (p) − f (Φ t (p))|/b, and f is decreasing along the flow Φ t . Now we are ready to prove Proposition 7.4 Proof. For any φ ∈ Ω 2. If z / ∈ supp(φ) ∪B D (p 0 ), y ∈ supp(φ) ∩B D are critical points off , and W s f (z) ∩ W ū f (y) = ∅, then W ū f (z) ∩ supp(φ) = ∅. This is because, by definition of D, Claim in Lemma 5.5 and properties of unstable As a result, by Proposition 7.3 φ (By Claim 1).

An counterexample
On closing this subsection, let's give a counterexample that when we drop the condition that ∇f has a positive lower bound near infinity,∂ 2 = 0 may fail. Consider the following heart shaped topological sphere S with obvious height function f . Then we have four critical points x, y, z, w as indicated below. Let γ be a flow line connecting y and w, and remove a point p on γ.
Recall that we assume in a neighborhood U x of critical points x of f , we have coordinate system z = (z 1 , ..., z n ), such that Moreover U x is an Euclidean open ball around x with radius 1. Also, these open balls are disjoint. We have the following observation: Lemma 7.8. V 0 can be written as disjoint union of ∪ x∈Crit(f ),n f (x)=0Ũx and V , where V is some open subset diffeomorphic to U c ,Ũ x is an Euclidean ball around x with radius 1 2 . V n is diffeomorphic to M.
Proof. Let X f := ∇f |∇f | 2 , Φ t be the flow generated by X f . Then we have This is because: Similarly, we can prove that V n is diffeomorphic to M.
Let C * (V i , U c ) be complex of relative singular chains. Then we have By a similar spectral sequence argument as in the proof of Theorem 1.6 in [2] and Lemma 7.8, one can show that Thus, it follows from the universal coefficient theorem that We will first show that the chain map J : (F in Section 5 is in fact an isomorphism when T is sufficiently large. Hence J induced an isomorphism between H * (2) (M, d T f ) and H * (C • (W u ), ∂) in that case.
More precisely we will follow the arguments in Chapter 6 of [21], with necessary modification, to show that there exists T 6 > T 0 , such that J is an isomorphism whenever T > T 6 . (We point out that the explicit description of T 6 is more involved than T 0 .) In fact, the only difference is that we need a more refined estimate in Theorem 6.17 of [21], that is: where P is the orthogonal projection from L 2 Λ(M ) to F [0,1], * , and C, c are positive constants.
Here τ x,T is defined as follows. Notice that in Section 4, we require that in a neighborhoof U of x, the metric and Morse function is of the form (4.1). Hence, let α x be a bump function whose support is contained in U , and α x ≡ 1 in a neighborhood V of x. Now let To obtain the estimate (7.4), pick a bump function η with compact support, such that η ≡ 1 on K. Then by our Agmon estimate, we have By Lemma 7.2, |α| ≤ C exp(−1/3ρ 7 ). Consequently, exp(−f )α ∈ L 2 Λ * (M ), and w = d 8f exp(−f )α is exact.

L is bijective
• L is injective: Next, choose a smooth function η, s.t. η = 0 on K, and η = 1 in f −1 ((S + 1, ∞) ∪ (−∞, −I − 1)). Then we set Thus we can see that for p ∈ U ′ c ′ , As a consequence, 2) , d T f ). Now we prove the claim: , and endow they with induced metrics. Define a diffeomorphism Ψ c ′ : Similarly, we can define a diffeomorphism Ψ ′ c : Then, where a ′ is some positive number which is smaller than a. (7.6)).

For the same reason, we have
• L is surjective: We claim that any cohomology class ξ ∈ H j (M, U c ) can be represented by a smooth closed j-form φ so that φ| Uc = 0. Also, it behaves on U as follows: ι X f φ = 0 and (Φ t ) * φ does not depend on t for large t. Then it follows that exp(−T f )φ belongs to L 2 Λ * (M ) (follows from the similar argument as above). Let ν ∈ Ker T f , s.t. ν − exp(−T f )φ is exact. Then we can see that L(ν) ∈ ξ hence L is surjective. This is because, we can find ψ ∈ Im(δ T f ), s.t.

Appendix: Decomposition of L 2 space
In this section, we investigate the decomposition (1.1). For this purpose we first have to understand the Friedrichs extension of ∆ H,f . Moreover, all operators considered in this section are closurable. Let V 1 to be the completion of V under · V 1 . Then for any β ∈ H, one can construct a bounded linear functional L β on V 1 as follows  and W be the completion of V under the norm · W . Then we can extend T toT min with Dom(T min ) = W. LetS max be the closure of S with Dom(S max ) = {α ∈ H : |(α, T φ) H | ≤ M α φ H , ∀φ ∈ V }. Namely, for any α ∈ Dom(S max ), since V is dense in H, by Riesz representation, there exists unique ν ∈ H, such that (ν, φ) H = (α, T φ). Now defineS max (α) = ν.

Review on Friedrichs extension
Since T V ⊂ V , S T is symmetric and nonnegative with Dom(ST ) = V.
Proposition 8.2. The Friedrichs extension ∆ of S T is justS maxTmin .
Proof. Since T V ⊂ V, we see that V 1 constructed in (8.1) is the same as W. Indeed, for any φ, ψ ∈ V, we have We now divide our discussion in two cases. (a) We first prove that DomS maxTmin ⊂ Dom(∆), and ∀α ∈ Dom(S max ), S maxTmin α = ∆α.