L^{2} harmonics forms on non compact manifolds.

# L2 harmonics forms on non compact manifolds.

Gilles Carron Departement de Mathematiques
Universite de Nantes
2 rue de la Houssiniere, BP 92208
44322 Nantes Cedex 03, France

The source of these notes is a series of lectures given at the CIMPA’s summer school ”Recent Topics in Geometric Analysis”. I want to thank the organizers of this summer school : Ahmad El Soufi and Mehrdad Shahsahani and I also want to thank Mohsen Rahpeyma who solved many delicate problems.

Theses notes aimed to give an insight into some links between cohomology, harmonics forms, the topology and the geometry of complete Riemannian manifolds. This is not a survey but a choice of few topics in a very large subject.

The first part can be regard as an introduction ; we define the space of harmonics forms, of cohomology. We recall the theorems of Hodge and de Rham on compact Riemannian manifolds. However the reader is assumed to be familiar with the basic of Riemannian geometry and with Hodge theory.

According to J. Roe ([55]) and following the classification of Von Neumann algebra, we can classify problems on harmonics forms in three types. The first one (type I) is the case where the space of harmonics forms has finite dimension, this situation is the nearest to the case of compact manifolds. The second (type II) is the case where the space of harmonics forms has infinite dimension but where we have a ”renormalized” dimension for instance when a discrete group acts cocompactly by isometry on the manifold; a good reference is the book of W. Lueck ([47]) and the seminal paper of M. Atiyah ([4]). The third type (type III) is the case where no renormalization procedure is available to define a kind of dimension of the space harmonics forms. Here we consider only the type I problems and at the end of the first part, we will prove a result of J. Lott which says that the finiteness of the dimension of the space of harmonics forms depends only on the geometry at infinity.

Many aspects111almost all in fact ! of harmonics forms will not be treated here : for instance we will not describe the important problem of the cohomology of locally symmetric spaces, and also we will not speak on the pseudo differential approach developped by R. Melrose and his school. However the reader will find at the end of this first chapter a list of some interesting results on the topological interpretation of the space of harmonics forms.

In the second chapter, we are interested in the space of harmonic forms. This space contains the differential of harmonic functions with gradient. We will not speak of the endpoint result of A. Grigory’an ([32, 31]) but we have include a study of P.Li and L-F. Tam ([39]) and of A. Ancona ([1]) on non parabolic ends. In this chapter, we will also study the case of Riemannian surfaces where this space depends only on the complex structure.

The last chapter focuses on the cohomology of conformally compact manifolds. The result is due to R. Mazzeo ([48]) and the proof present here is the one of N. Yeganefar ([65]) who used an integration by parts formula due to H.Donnelly and F.Xavier ([24]).

## 1. A short introduction to L2 cohomology

In this first chapter, we introduce the main definitions and prove some preliminary results.

### 1.1. Hodge and de Rham ’s theorems

#### 1.1.1. de Rham ’s theorem

Let be a smooth manifold of dimension , we denote by the space of smooth differential forms on and by the subspace of formed by forms with compact support; in local coordinates , an element has the following expression

 α=∑I={i1

where are smooth functions of . The exterior differentiation is a differential operator

 d:C∞(ΛkT∗M)→C∞(Λk+1T∗M),

locally we have

 d(∑IαIdxI)=∑IdαI∧dxI.

This operator satisfies , hence the range of is included in the kernel of .

###### Definition 1.1.

The de Rham’s cohomology group of is defined by

 HkdR(M)={α∈C∞(ΛkT∗M),dα=0}dC∞(Λk−1T∗M).

These spaces are clearly diffeomorphism invariants of , moreover the deep theorem of G. de Rham says that these spaces are isomorphic to the real cohomology group of , there are in fact homotopy invariant of :

###### Theorem 1.2.
 HkdR(M)≃Hk(M,R).

From now, we will suppress the subscript for the de Rham’s cohomology. We can also define the de Rham’s cohomology with compact support.

###### Definition 1.3.

The de Rham’s cohomology group with compact support of is defined by

 Hk0(M)={α∈C∞0(ΛkT∗M),dα=0}dC∞0(Λk−1T∗M).

These spaces are also isomorphic to the real cohomology group of with compact support. When is the interior of a compact manifold with compact boundary

 M=¯¯¯¯¯¯M∖∂¯¯¯¯¯¯M,

then is isomorphic to the relative cohomology group of :

 Hk0(M)=Hk(¯¯¯¯¯¯M,∂¯¯¯¯¯¯M):={α∈C∞(ΛkT∗¯¯¯¯¯¯M),dα=0,ι∗α=0}{dβ,β∈C∞(Λk−1T∗¯¯¯¯¯¯M) and ι∗β=0}

where is the inclusion map.

#### 1.1.2. Poincaré duality

When we assume that is oriented 222It is not a serious restriction we can used cohomology with coefficient in the orientation bundle. the bilinear map

 Hk(M)×Hn−k0(M)→R([α],[β])↦∫Mα∧β:=I([α],[β])

is well defined, that is to say doesn’t depend on the choice of representatives in the cohomology classes or (this is an easy application of the Stokes formula). Moreover this bilinear form provides an isomorphism between and . In particular when is closed () and satisfies that

 ∀[β]∈Hn−k0(M), ∫Mα∧β=0

then there exists such that

 α=dγ.

#### 1.1.3. L2 cohomology

1.1.3 .a) The operator . We assume now that is endowed with a Riemannian metric , we can define the space whose elements have locally the following expression

 α=∑I={i1

where and globally we have

 ∥α∥2L2:=∫M|α(x)|2g(x)dvolg(x)<∞.

The space is a Hilbert space with scalar product :

 ⟨α,β⟩=∫M(α(x),β(x))g(x)dvolg(x).

We define the formal adjoint of :

 d∗:C∞(Λk+1T∗M)→C∞(ΛkT∗M)

by the formula

 ∀α∈C∞0(Λk+1T∗M) and β∈C∞0(ΛkT∗M),            ⟨d∗α,β⟩=⟨α,dβ⟩ .

When is the Levi-Civita connexion of , we can give local expressions for the operators and : let be a local orthonormal frame and let be its dual frame :

 θi(X)=g(Ei,X)

then

 (1.1) dα=n∑i=1θi∧∇Eiα,

and

 (1.2) d∗α=−n∑i=1intEi(∇Eiα),

where we have denote by the interior product with the vector field .
1.1.3 .b) harmonic forms. We consider the space of closed forms :

 Zk2(M)={α∈L2(ΛkT∗M), dα=0}

where it is understood that the equation holds weakly that is to say

 ∀β∈C∞0(Λk+1T∗M), ⟨α,d∗β⟩=0.

That is we have :

 Zk2(M)=(d∗C∞(Λk+1T∗M))⊥,

hence is a closed subspace of . We can also define

 Hk(M)=(d∗C∞0(Λk+1T∗M))⊥∩(dC∞0(Λk−1T∗M))⊥=Zk2(M)∩{α∈L2(ΛkT∗M), d∗α=0}={α∈L2(ΛkT∗M), dα=0 and d∗α=0}.

Because the operator is elliptic, we have by elliptic regularity : . We also remark that by definition we have

 ∀α∈C∞0(Λk−1T∗M), ∀β∈C∞0(Λk+1T∗M)⟨dα,d∗β⟩=⟨ddα,β⟩=0

Hence

 dC∞0(Λk−1T∗M)⊥d∗C∞0(Λk+1T∗M)

and we get the Hodge-de Rham decomposition of

 (1.3) L2(ΛkT∗M)=Hk(M)⊕¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯dC∞0(Λk−1T∗M)⊕¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯d∗C∞0(Λk+1T∗M) ,

where the closures are taken for the topology. And also

 (1.4) Hk(M)≃Zk2(M)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯dC∞0(Λk−1T∗M).

1.1.3 .c) cohomology: We also define the (maximal) domain of by

 Dk(d)={α∈L2(ΛkT∗M), dα∈L2}

that is to say if and only if there is a constant such that

 ∀β∈C∞0(Λk+1T∗M), |⟨α,d∗β⟩|≤C∥β∥2.

In that case, the linear form extends continuously to and there is such that

 ∀β∈C∞0(Λk+1T∗M), ⟨α,d∗β⟩=⟨γ,β⟩.

We remark that we always have .

###### Definition 1.4.

We define the space of reduced cohomology by

 Hk2(M)=Zk2(M)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯dDk−1(d).

The space of non reduced cohomology is defined by

 nrHk2(M)=Zk2(M)dDk−1(d).

These two spaces coincide when the range of is closed; the first space is always a Hilbert space and the second is not necessary Hausdorff. We also have hence we always get a surjective map :

 Hk(M)→Hk2(M)→{0}.

In particular any class of reduced cohomology contains a smooth representative.
1.1.3 .d) Case of complete manifolds. The following result is due to Gaffney ([28], see also part 5 in [64]) for a related result)

###### Lemma 1.5.

Assume that is a complete Riemannian metric then

 ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯dDk−1(d)=¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯dC∞0(Λk−1T∗M).
###### Proof.

We already know that , moreover using a partition of unity and local convolution it is not hard to check that if has compact support then we can find a sequence of smooth forms with compact support such that

 ∥αl−α∥L2+∥dαl−dα∥L2≤1/l.

So we must only prove that if then we can build a sequence of elements of with compact support such that

 L2−limN→∞dαN=dα.

We fix now an origin and denote by the closed geodesic ball of radius and centered at , because is assumed to be complete we know that is compact and

 M=∪N∈NB(o,N).

We consider with with support in such that

 ρ=1 on [0,1/2]

and we define

 (1.5) χN(x)=ρ(d(o,x)N).

Then is a Lipschitz function and is differentiable almost everywhere and

 dχN(x)=ρ′(d(o,x)N)dr

where is the differential of the function . Let and define

 αN=χNα,

the support of is included in the ball of radius and centered at hence is compact. Moreover we have

 ∥αN−α∥L2≤∥α∥L2(M∖B(o,N/2))

hence

 L2−limN→∞αN=α

Moreover when we have

 ⟨αN,d∗φ⟩=⟨α,χNd∗φ⟩=⟨α,d∗(χNφ)⟩+⟨α,int−−−→gradχNφ⟩=⟨χNdα+dχN∧α,φ⟩

Hence and

 dαN=χNdα+dχN∧α.

But for almost all , we have hence

 ∥dαN−dα∥L2≤∥ρ′∥L∞N∥α∥L2+∥χNdα−dα∥L2≤∥ρ′∥L∞N∥α∥L2+∥dα∥L2(M∖B(o,N/2)) .

Hence we have build a sequence of elements of with compact support such that

A corollary of this lemma (1.5) and of (1.4) is the following :

###### Corollary 1.6.

When is a complete Riemannian manifold then the space of harmonic forms computes the reduced cohomology :

 Hk2(M)≃Hk(M).

With a similar proof, we have another result :

###### Proposition 1.7.

When is a complete Riemannian manifold then

 Hk(M)={α∈L2(ΛkT∗M),(dd∗+d∗d)α=0}.
###### Proof.

Clearly we only need to check the inclusion :

 {α∈L2(ΛkT∗M),(dd∗+d∗d)α=0}⊂Hk(M).

We consider again the sequence of cut-off functions defined previously in (1.5). Let satisfying by elliptic regularity we know that is smooth. Moreover we have :

 ∥d(χNα)∥2L2=∫M[|dχN∧α|2+2⟨dχN∧α,χNdα⟩+χ2N|dα|2]dvolg=∫M[|dχN∧α|2+⟨dχ2N∧α,dα⟩+χ2N|dα|2]dvolg=∫M[|dχN∧α|2+⟨d(χ2Nα),dα⟩]dvolg=∫M[|dχN∧α|2+⟨χ2Nα,d∗dα⟩]dvolg

Similarly we get :

 ∥d∗(χNα)∥2L2=∫M[|int−−−→gradχNα|2+⟨χ2Nα,dd∗α⟩]dvolg

Summing these two equalities we obtain :

 ∥d∗(χNα)∥2L2+∥d∗(χNα)∥2L2=∫M|dχN|2|α|2dvolg≤∥ρ′∥2L∞N2∫M|α|2dvolg.

Hence when tends to we obtain

 ∥d∗α∥2L2+∥d∗α∥2L2=0.

This proposition has the consequence that on a complete Riemannian manifold harmonic functions are closed hence locally constant. Another corollary is that the reduced cohomology of the Euclidean space is trivial333This can also be proved with the Fourier transform. :

###### Corollary 1.8.
 Hk2(Rn)={0}.
###### Proof.

On the Euclidean space a smooth form can be expressed as

 α=∑I={i1

and will be a solution of the equation

 (dd∗+d∗d)α=0

if and only if all the functions are harmonic and hence zero because the volume of is infinite.∎

###### Remark 1.9.

When is not complete, we have not necessary equality between the space (whose elements are sometimes called harmonics fields) and the space of the solutions of the equation . For instance, on the interval the space is the space of constant functions, whereas solutions of the equation are affine. More generally, on a smooth compact connected manifold with smooth boundary endowed with a smooth Riemannian metric, then again is the space of constant functions, whereas the space is the space of harmonic function; this space is infinite dimensionnal when .

1.1.3e) Case of compact manifolds The Hodge’s theorem says that for compact manifold cohomology is computed with harmonic forms :

###### Theorem 1.10.

If is a compact Riemannian manifold without boundary then

 Hk2(M)≃Hk(M)≃Hk(M).

When is the interior of a compact manifold with compact boundary and when extends to (hence is incomplete) a theorem of P. Conner ([19]) states that

 Hk2(M)≃Hk(M)≃Hkabs(¯¯¯¯¯¯M)

where

 Hkabs(M)={α∈L2(ΛkT∗M), dα=d∗α=0 and int→να=0 along ∂¯¯¯¯¯¯M}

and is the inward unit normal vector field. In fact when is a compact subset of with smooth boundary and if is a complete Riemannian metric on then for , we also have the equality

 Hk2(Ω)≃Hkabs(Ω)

where if is the inward unit normal vector field, we have also denoted

 (1.6) Hkabs(Ω)={α∈L2(ΛkT∗Ω), dα=d∗α=0 and int→να=0 % along ∂Ω}.

### 1.2. Some general properties of reduced L2 cohomology

#### 1.2.1. a general link with de Rham’s cohomology

We assume that is a complete Riemannian manifold, the following result is due to de Rham (theorem 24 in [21])

###### Lemma 1.11.

Let and suppose that is zero in that is there is a sequence such that

 α=L2−liml→∞dβl

then there is such that

 α=dβ.

In full generality, we know nothing about the behavior of at infinity.

###### Proof.

We can always assume that is oriented, hence by the Poincaré duality (1.1.2), we only need to show that if is closed then

 ∫Mα∧ψ=0 .

But by assumption,

 ∫Mα∧ψ=liml→∞∫Mdβl∧ψ=liml→∞∫Md(βl∧ψ) because ψ is closed=0.

Hence the result.∎

This lemma implies the following useful result which is due to M. Anderson ([2]):

###### Corollary 1.12.

There is a natural injective map

 Im(Hk0(M)→Hk(M))→Hk2(M).
###### Proof.

As a matter of fact we need to show that if is closed and zero in the reduced cohomology then it is zero in usual cohomology: this is exactly the statement of the previous lemma (1.11). ∎

#### 1.2.2. Consequence for surfaces.

These results have some implications for a complete Riemannian surface :

• If the genus of is infinite then the dimension of the space of harmonic forms is infinite.

• If the space of harmonics forms is trivial then the genus of is zero and is diffeomorphic to a open set of the sphere.

As a matter of fact, a handle of is a embedding such that if we denote then is connected.

We consider now a function on depending only on the second variable such that

 ρ(θ,t)={1 when t>1/20 when t<−1/2

then is a form with compact support in and we can extend to all ; we obtain a closed -form also denoted by which has compact support in . Moreover because is connected, we can find a continuous path joining to ; we can defined the loop given by

 γ(t)={f(1,t) for t∈[−1,1]c(t−1) for t∈[1,2].

It is easy to check that

 ∫γα=1 ,

hence is not zero in . A little elaboration from this argument shows that

 genus(S)≤dimIm(H10(M)→H1(M)).

### 1.3. Lott’s result

We will now prove the following result due to J.Lott ([44]):

###### Theorem 1.13.

Assume that and are complete oriented manifold of dimension which are isometric at infinity that is to say there are compact sets and such that and are isometric. Then for

 dimHk(M1,g1)<∞⇔dimHk(M2,g2)<∞.

We will give below the proof of this result, this proof contains many arguments which will be used and refined in the next two lectures. In view of the Hodge-de Rham theorem and of J. Lott’s result, we can ask the following very general questions :

1. What are the geometry at infinity iensuring the finiteness of the dimension of the spaces ?

Within a class of Riemannian manifold having the same geometry at infinity :

2. What are the links of the spaces of reduced cohomology with the topology of and with the geometry ”at infinity” of ?

There is a lot of articles dealing with these questions, I mention some of them :

1. In the pioneering article of Atiyah-Patodi-Singer ([5]), the authors considered manifold with cylindrical end : that is to say there is a compact of such that is isometric to the Riemannian product . Then they show that the dimension of the space of -harmonic forms is finite ; and that these spaces are isomorphic to the image of the relative cohomology in the absolute cohomology. These results were used by Atiyah-Patodi-Singer in order to obtain a formula for the signature of compact Manifolds with boundary.

2. In [48, 50], R. Mazzeo and R.Phillips give a cohomological interpretation of the space for geometrically finite real hyperbolic manifolds. These manifolds can be compactified. They identify the reduced cohomology with the cohomology of smooth differential forms satisfying certain boundary conditions.

3. The solution of the Zucker’s conjecture by L.Saper-M.Stern and E.Looijenga ([46],[58]) shows that the spaces of harmonic forms on Hermitian locally symmetric space with finite volume are isomorphic to the middle intersection cohomology of the Baily-Borel-Satake compactification of the manifold. An extension of this result has been given by A.Nair and L.Saper ([52],[56]). Moreover recently, L. Saper obtains the topological interpretation of the reduced cohomology of any locally symmetric space with finite volume ([57]). In that case the finiteness of the dimension of the space of harmonics forms is due to A. Borel and H.Garland ([8]).

4. According to Vesentini ([63]) if is flat outside a compact set, the spaces are finite dimensional. J. Dodziuk asked about the topological interpretation of the space ([22]). In this case, the answer has been given in [15].

5. In a recent paper ([33]) Tamás Hausel, Eugenie Hunsicker and Rafe Mazzeo obtain a topological interpretation of the cohomology of complete Riemannian manifold whose geometry at infinity is fibred boundary and fibred cusp (see [49, 62]). These results have important application concerning the Sen’s conjecture ([34, 59]).

6. In [45], J. Lott has shown that on a complete Riemannian manifold with finite volume and pinched negative curvature, the space of harmonic forms has finite dimension. N. Yeganefar obtains the topological interpretation of these spaces in two cases, first when the curvature is enough pinched ([65]) and secondly when the metric is Kähler ([66]).

Proof of J. Lott’s result. We consider a complete oriented Riemannian manifold and a compact subset with smooth boundary and we let be the exterior of , we are going to prove that

 dimHk2(M)<∞⇔dimHk2(Ω)<∞ ;

this result clearly implies Lott’s result.

The co boundary map is defined as follow: let where is a smooth representative of , we choose a smooth extension of , then is a closed smooth form with support in and if is the inclusion we have . Some standard verifications show that

 b(c)=[d¯α]∈Hk+1(K,∂K)

is well defined, that is it doesn’t depend of the choice of nor on the smooth extension of .

The inclusion map induced a linear map (the restriction map)

 [j∗Ω]:Hk2(M)→Hk2(Ω).
###### Lemma 1.14.

We always have

 kerb=Im[j∗Ω].

Proof of lemma 1.14. First by construction we have hence we only need to prove that

 kerb⊂Im[j∗Ω].

Let and let a smooth representative of , we know that has a smooth extension such that is zero in . That is to say there is a smooth form such that

 d¯α=dβ on K and ι∗β=0

We claim that the form defined by

 ~α={¯α−β on Kα on Ω

is weakly closed. As a matter of fact, we note the unit normal vector field pointing into and the inclusion, then let with the Green’s formula, we obtain

 ∫M(~α,d∗φ)=∫K(¯α−β,d∗φ)+∫Ω(α,d∗φ)=−∫∂K(ι∗(¯α−β),int→νφ)dσ+∫∂K(ι∗¯α,int→νφ)dσ=0.

We clearly have , hence

Now because has finite dimension, we know that

 dimIm[j∗Ω]<∞⇔dimHk2(Ω)<∞.

Hence we get the implication :

 dimHk2(M)<∞⇒dimHk2(Ω)<∞.

To prove the reverse implication, we consider the reduced cohomology of relative to the boundary . We introduce

We remark here that the elements of have compact support in in particular their support can touch the boundary; in fact a smooth closed form belongs to if and only its pull-back by is zero. This is a consequence of the integration by part formula

 ∫Ω(α,d∗φ)dvolg=∫Ω(dα,φ)dvolg+∫∂Ω(ι∗α,int→νφ)dσ

where is the Riemannian volume on induced by the metric and is the unit inward normal vector field.

We certainly have and we define

 Hk2(Ω,∂Ω)=Zk2(Ω,∂Ω)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯dC∞0(ΛkT∗Ω) .

These relative (reduced) cohomology space can be defined for every Riemannian manifold with boundary. In fact, these relative (reduced) cohomology spaces also have an interpretation in terms of harmonics forms :

 (1.7) Hk2(Ω,∂Ω)≃Hkrel(Ω)

where

 Hkrel(Ω)={α∈L2(ΛkT∗Ω),dα=d∗α=0 and ι∗α=0}.

There is a natural map : the extension by zero map :

 e:Hk2(Ω,∂Ω)→Hk2(M).

When we define to be on and zero on , is clearly a bounded map moreover

 e(dC∞0(ΛkT∗Ω))⊂dC∞0(Λ