HARMONICS FORMS ON NON COMPACT MANIFOLDS

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2L HARMONICS FORMS ON NON COMPACT MANIFOLDS.GILLES CARRONThe source of these notes is a series of lectures given at the CIMPA’s summerschool ”Recent Topics in Geometric Analysis”. I want to thank the organizers ofthis summer school : Ahmad El Soufi and Mehrdad Shahsahani and I also want tothank Mohsen Rahpeyma who solved many delicate problems.2Theses notes aimed to give an insight into some links between L cohomology,2L harmonics forms, the topology and the geometry of complete Riemannian man-ifolds. This is not a survey but a choice of few topics in a very large subject.2The first part can be regard as an introduction ; we define the space of L2harmonics forms, of L cohomology. We recall the theorems of Hodge and deRham on compact Riemannian manifolds. However the reader is assumed to befamiliar with the basic of Riemannian geometry and with Hodge theory.According to J. Roe ([55]) and following the classification of Von Neumann alge-2bra, we can classify problems on L harmonics forms in three types. The first one2(type I) is the case where the space of harmonics L forms has finite dimension,this situation is the nearest to the case of compact manifolds. The second (type2II) is the case where the space of harmonics L forms has infinite dimension butwhere we have a ”renormalized” dimension for instance when a discrete group actscocompactlybyisometryonthemanifold; agoodreferenceisthebookofW.Lueck([47]) and the seminal paper of M. Atiyah ([4]). The third type ...

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L 2 HARMONICS FORMS ON NON COMPACT MANIFOLDS.

GILLES CARRON

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 betweenL 2 cohomology, L 2 topology and the geometry of complete Riemannian man-harmonics forms, the ifolds. 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 ofL 2 harmonics forms, ofL 2 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 alge- bra, we can classify problems onL 2 harmonics forms in three types. The first one (type I) is the case where the space of harmonicsL 2 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 harmonicsL 2 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 spaceL 2  we consider only the type I problems and Hereharmonics forms. 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 ofL 2 harmonics forms depends only on the geometry at infinity. Many aspects 1 ofL 2 harmonics  forforms will not be treated here : instance we will not describe the important problem of theL 2 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 ofL 2 harmonics forms. In the second chapter, we are interested in the space of harmonicL 2 1−forms. This space contains the differential of harmonic functions withL 2 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 theL 2 cohomology of conformally compact mani- folds. The result is due to R. Mazzeo ([48]) and the proof present here is the one 1 almost all in fact !

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2 GILLES CARRON of N. Yeganefar ([65]) who used an integration by parts formula due to H.Donnelly and F.Xavier ([24]). Contents 1. A short introduction toL 2 cohomology 2 1.1. Hodge and de Rham ’s theorems 2 1.2. Some general properties of reducedL 2 cohomology 8 1.3. Lott’s result 10 1.4. Some bibliographical hints 14 2. HarmonicsL 2 1−forms 14 2.1. Ends 14 2.2.H 01 (M) versusH 21 (M) 16 2.3. The two dimensional case 26 2.4. Bibliographical hints 29 3.L 2  29cohomology of conformally compact manifold 3.1. The geometric setting 29 3.2. The case wherek= dimM/2. 30 3.3. A reduction to exact metric 31 3.4. A Rellich type identity and its applications 32 3.5. Application to conformally compact Riemannian manifold 33 3.6. The spectrum of the Hodge-deRham Laplacian 34 3.7. Applications to conformally compact manifolds 36 3.8. Mazzeo’s result. 39 3.9. Bibliographical hints 40 References 40

1.A short introduction toL 2 cohomology In this first chapter, we introduce the main definitions and prove some prelimi- nary results. 1.1.Hodge and de Rham ’s theorems. 1.1.1.de Rham ’s theorem.LetM n be a smooth manifold of dimensionn, we denote byC ∞ (Λ k T ∗ M) the space of smooth differentialk−forms onMand by C 0 ∞ (Λ k T ∗ M) the subspace ofC ∞ (Λ k T ∗ M) formed by forms with compact support; in local coordinates (x 1 , x 2 , ..., x n ), an elementα∈C ∞ (Λ k T ∗ M) has the following expression α= X α I dx i 1 ∧dx i 2 ∧...∧dx i k = X α I dx I I={i 1 <i 2 <...<i k } I whereα I are smooth functions of (x 1 , x 2 , ..., x n  exterior differentiation is a). The differential operator d:C ∞ (Λ k T ∗ M)→C ∞ (Λ k+1 T ∗ M), locally we have d
X I α I dx I ! = I X dα I ∧dx I .

L 2  3HARMONICS FORMS ON NON COMPACT MANIFOLDS. This operator satisfiesd◦d= 0, hence the range ofdis included in the kernel ofd. Definition1.1.Thek th de Rham’s cohomology group ofMis defined by H k Rd (M) =  α∈C dC ∞ ∞ Λ (Λ ( k k T − ∗ 1 T M ∗ ) M α,d ) 0=  . These spaces are clearly diffeomorphism invariants ofM, moreover the deep theorem of G. de Rham says that these spaces are isomorphic to the real cohomology group ofM, there are in fact homotopy invariant ofM: Theorem 1.2. k H dR (M)'H k (M,R). From now, we will suppress the subscriptdRforthedeRham’scohmologo.yeW can also define the de Rham’s cohomology with compact support. Definition1.3.Thek th de Rham’s cohomology group with compact support ofM is defined by ∈C 0 ∞ (Λ k T ∗ M), dα= 0  H 0 k (M) =  α dC 0 ∞ (Λ k−1 T ∗ M) . These spaces are also isomorphic to the real cohomology group ofMwith com- pact support. WhenMis the interior of a compact manifoldMwith compact boundary∂M M=M\∂M , thenH 0 k (M) is isomorphic to the relative cohomology group ofM: H 0 k (M) =H k (M , ∂M) :=   ,βd α∈ β C ∈ ∞ C ( ∞ Λ k ( T Λ k ∗ − M 1 T ), ∗ d M α n)a 0=, d ι ∗ ι ∗ β α0 = = 0   whereι:∂M→Mis the inclusion map. 1.1.2.cnioe´raPalduy.itWhen we assume thatMis oriented 2 the bilinear map H k (M)×H 0 n−k (M)→R ([α],[β])7→ Z α∧β:=I([α],[β]) M is well defined, that is to sayI([α],[β]) doesn’t depend on the choice of representa- tives in the cohomology classes [α] or [β] (this is an easy application of the Stokes formula). Moreover this bilinear form provides an isomorphism betweenH k (M) and  H 0 n−k (M)  ∗ . In particular whenα∈C ∞ (Λ k T ∗ M) is closed (dα= 0) and satisfies that ∀[β]∈H 0 n−k (M), Z α∧β= 0 M then there existsγ∈C ∞ (Λ k−1 T ∗ M) such that α=dγ. 2 It is not a serious restriction we can used cohomology with coefficient in the orientation bundle.

4 GILLES CARRON 1.1.3.L 2 cohomology.1.1.3.a) The operatord ∗ .We assume now thatMis en- dowed with a Riemannian metricg, we can define the spaceL 2 (Λ k T ∗ M) whose elements have locally the following expression α= X α I dx i 1 ∧dx i 2 ∧...∧dx i k I={i 1 <i 2 <...<i k } whereα I ∈L l 2 oc and globally we have kαk 2 L2 := Z M |α(x)| 2 g(x) dvol g (x)<∞. The spaceL 2 (Λ k T ∗ M) is a Hilbert space with scalar product : hα, βi= Z M (α(x), β(x)) g(x) dvol g (x). We define the formal adjoint ofd: d ∗ :C ∞ (Λ k+1 T ∗ M)→C ∞ (Λ k T ∗ M) by the formula ∀α∈C 0 ∞ (Λ k+1 T ∗ M) andβ∈C 0 ∞ (Λ k T ∗ M), hd ∗ α, βi=hα, dβi. Whenris the Levi-Civita connexion ofg, we can give local expressions for the operatorsdandd ∗ : let (E 1 , E 2 , ..., E n ) be a local orthonormal frame and let (θ 1 , θ 2 , ..., θ n ) be its dual frame : θ i (X) =g(E i , X) then n (1.1)dα= X θ i ∧ r E i α, i=1 and n (1.2)d ∗ α=− X int E i (r E i α), i=1 where we have denote by int E i the interior product with the vector fieldE i . 1.1.3.b)L 2 harmonic forms.We consider the space ofL 2 closed forms : Z k (M) ={α∈L 2 (Λ k T ∗ M), dα= 0} 2 where it is understood that the equationdα= 0 holds weakly that is to say ∀β∈C 0 ∞ (Λ k+1 T ∗ M),hα, d ∗ βi= 0. That is we have : Z k (M) =  d ∗ C ∞ (Λ k+1 T ∗ M)  ⊥ , 2 henceZ 2 k (M) is a closed subspace ofL 2 (Λ k T ∗ M). We can also define H k (M) =  d ∗ C 0 ∞ (Λ k+1 T ∗ M)  ⊥ ∩  dC 0 ∞ (Λ k−1 T ∗ M)  ⊥ 2 =Z 2 k (M)∩ {α∈L(Λ k T ∗ M), d ∗ α= 0} ={α∈L 2 (Λ k T ∗ M), dα= 0 andd ∗ α= 0}.

L 2 HARMONICS FORMS ON NON COMPACT MANIFOLDS. 5 Because the operatord+d ∗ elliptic, we have by elliptic regularity :is H k (M)⊂ C ∞ (Λ k T ∗ M also remark that by definition we have). We ∀α∈C 0 ∞ (Λ k−1 T ∗ M),∀β∈C 0 ∞ (Λ k+1 T ∗ M) hdα, d ∗ βi=hddα, βi= 0 Hence dC 0 ∞ (Λ k−1 T ∗ M)⊥d ∗ C 0 ∞ (Λ k+1 T ∗ M) and we get the Hodge-de Rham decomposition ofL 2 (Λ k T ∗ M) (1.3)L 2 (Λ k T ∗ M) =H k (M)⊕ dC 0 ∞ (Λ k−1 T ∗ M)⊕d ∗ C 0 ∞ (Λ k+1 T ∗ M), where the closures are taken for theL 2  alsotopology. And ' (1.4)H k (M) dC 0 ∞ ( Z Λ 2 k k ( − M 1 ) T ∗ M) . 1.1.3.c)L 2 cohomology:We also define the (maximal) domain ofdby D k (d) ={α∈L 2 (Λ k T ∗ M), dα∈L 2 } that is to sayα∈ D k (d) if and only if there is a constantCsuch that ∀β∈C 0 ∞ (Λ k+1 T ∗ M),|hα, d ∗ βi| ≤Ckβk 2 . In that case, the linear formβ∈C 0 ∞ (Λ k+1 T ∗ M)7→ hα, d ∗ βiextends continuously toL 2 (Λ k+1 T ∗ M) and there isγ=:dαsuch that ∀β∈C 0 ∞ (Λ k+1 T ∗ M),hα, d ∗ βi=hγ, βi. We remark that we always havedD k−1 (d)⊂Z 2 k (M). Definition1.4.We define thek th space of reducedL 2 cohomology by H 2 k (M) = dD Z 2 k k ( − M 1 ) (d) . Thek th space of non reducedL 2 cohomology is defined by k nr H 2 (M) = Z dD 2 k k ( − M 1 ) (d) . These two spaces coincide when the range ofd:D k−1 (d)→L 2 is closed; the first space is always a Hilbert space and the second is not necessary Hausdorff. We also haveC 0 ∞ (Λ k−1 T ∗ M)⊂ D k−1 (d) hence we always get a surjective map : H k (M)→H 2 k (M)→ {0}. In particular any class of reducedL 2 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 thatgis a complete Riemannian metric then dD k−1 (d) =dC 0 ∞ (Λ k−1 T ∗ M).

6 GILLES CARRON Proof.We already know thatdC 0 ∞ (Λ k−1 T ∗ M)⊂ D k−1 (d), moreover using a par- tition of unity and local convolution it is not hard to check that ifα∈ D k−1 (d) has compact support then we can find a sequence (α l ) l∈N of smooth forms with compact support such that kα l −αk L2 +kdα l −dαk L2 ≤1/l. So we must only prove that ifα∈ D k−1 (d) then we can build a sequence (α N ) N of elements ofD k−1 (d) with compact support such that L 2 −limdα N =dα. N→∞ We fix now an origino∈Mand denote byB(o, N) the closed geodesic ball of radiusNand centered ato, because (M, g) is assumed to be complete we know thatB(o, N) is compact and M=∪ N∈N B(o, N). We considerρ∈C 0 ∞ (R + ) with 0≤ρ≤1 with support in [0,1] such that ρ= 1 on [0,1/2] and we define (1.5)χ N (x) =ρ  d(o,x N )  . Thenχ N is a Lipschitz function and is differentiable almost everywhere and dχ N (x) =ρ 0  d( N ,xo)  dr wheredris the differential of the functionx7→d(o, x). Letα∈ D k−1 (d) and define α N =χ N α, the support ofα N is included in the ball of radiusNand centered atohence is compact. Moreover we have kα N −αk L2 ≤ kαk L2(M\B(o,N/2)) hence L 2 −limα N =α N→∞ Moreover whenϕ∈C 0 ∞ (Λ k+1 T ∗ M) we have hα N , d ∗ ϕi=hα, χ N d ∗ ϕi =hα, d ∗ (χ N ϕ)i+hα,int g − r − a → d χ N ϕi =hχ N dα+dχ N ∧α, ϕi Henceα N ∈ D k−1 (d) and dα N =χ N dα+dχ N ∧α. But for almost allx∈M, we have|dχ N |(x)≤ kρ 0 k L∞ /Nhence kdα N −dαk L2 ≤ kρ 0 N k L∞ kαk L2 +kχ N dα−dαk L2 L∞ ≤ kρ 0 N k k αk L2 +kdαk L2(M\B(o,N/2)) .

L 2 HARMONICS FORMS ON NON COMPACT MANIFOLDS. 7 Hence we have build a sequenceα N of elements ofD k−1 (d) with compact support such thatL 2 −lim N→∞ dα N =dα. A corollary of this lemma (1.5) and of (1.4) is the following : Corollary 1.6.When(M, g)is a complete Riemannian manifold then the space of harmonicL 2 forms computes the reducedL 2 cohomology : H 2 k (M)' H k (M). With a similar proof, we have another result : Proposition 1.7.When(M, g)is a complete Riemannian manifold then H k (M) ={α∈L 2 (Λ k T ∗ M),(dd ∗ +d ∗ d)α= 0}. Proof.Clearly we only need to check the inclusion : {α∈L 2 (Λ k T ∗ M),(dd ∗ +d ∗ d)α= 0} ⊂ H k (M). We consider again the sequence of cut-off functionsχ N defined previously in (1.5). Letα∈L 2 (Λ k T ∗ M) satisfying (dd ∗ +d ∗ d)α= 0 by elliptic regularity we know that α Moreover we have :is smooth. kd(χ N α)k 2 L2 =  Z |dχ N ∧α| 2 + 2hdχ N ∧α, χ N dαi+χ 2 N |dα| 2  dvol g M = Z M  |dχ N ∧α| 2 +hdχ 2 N ∧α, dαi+χ 2 N |dα| 2  dvol g  |dχ N χ 2 N α), dαi  dvol g = Z M ∧α| 2 +hd( dχ N ∧α, d ∗ dαi  dvol g = Z M  |α| 2 +hχ 2 N Similarly we get : kd ∗ (χ N α)k 2 L2 = Z M h |int g − r − a → d χ N α| 2 +hχ 2 N α, dd ∗ αi i dvol g Summing these two equalities we obtain : 2 kd ∗ (χ N α)k 2 L2 +kd ∗ (χ N α)k 2 L2 = Z M |dχ N | 2 |α| 2 dvol g ≤ kρ 0 N k 2 Z M |α| 2 dvol g . L∞ Hence whenNtends to∞we obtain kd ∗ αk 2 L2 +kd ∗ αk 2 L2 = 0.  This proposition has the consequence that on a complete Riemannian manifold harmonicL 2 functions are closed hence locally constant. Another corollary is that the reducedL 2 cohomology of the Euclidean space is trivial 3 : Corollary 1.8. H 2 k (R n ) ={0}. 3 This can also be proved with the Fourier transform.

8 GILLES CARRON Proof.On the Euclidean spaceR n a smoothkformαcan be expressed as α= X α I dx i 1 ∧dx i 2 ∧...∧dx i k I={i 1 <i 2 <...<i k } andαwill be aL 2 solution of the equation (dd ∗ +d ∗ d)α= 0 if and only if all the functionsα I are harmonic andL 2 hence zero because the volume ofR n is infinite. Remark1.9.When (M, g) is not complete, we have not necessary equality between the spaceH k (M) (whose elements are sometimes called harmonics fields) and the space of theL 2 solutions of the equation (dd ∗ +d ∗ d)α instance, on the For= 0. intervalM= [0,1] the spaceH 0 (M) is the space of constant functions, whereas L 2 solutions of the equation (dd ∗ +d ∗ d)α generally, on a More= 0 are affine. smooth compact connected manifold with smooth boundary endowed with a smooth Riemannian metric, then againH 0 (M) is the space of constant functions, whereas the space{f∈L 2 (M), d ∗ df= 0}is the space of harmonicL 2 function; this space is infinite dimensionnal when dimM >1. 1.1.3e) Case of compact manifoldsThe Hodge’s theorem says that for compact manifold cohomology is computed with harmonic forms : Theorem 1.10.IfMis a compact Riemannian manifold without boundary then H 2 k (M)' H k (M)'H k (M). WhenMis the interior of a compact manifoldMwith compact boundary∂M and whengextends toM(hencegis incomplete) a theorem of P. Conner ([19]) states that H 2 k (M)'H k (M)' H bs k a (M) where H sab k (M) ={α∈L 2 (Λ k T ∗ M), dα=d ∗ α int= 0 and ~ν α= 0 along∂M} andν~:∂M→T Mis  Inthe inward unit normal vector field. fact whenK⊂Mis a compact subset ofMwith smooth boundary and ifgis a complete Riemannian metric onMthen for Ω =M\K, we also have the equality H 2 k (Ω)' H bs k a (Ω) where ifν~:∂Ω→T Mis the inward unit normal vector field, we have also denoted (1.6)H sab k (Ω) ={α∈L 2 (Λ k T ∗ Ω), dα=d ∗ α= 0 and int ~ν α= 0 along∂Ω}. 1.2.Some general properties of reducedL 2 cohomology. 1.2.1.a general link with de Rham’s cohomology.We assume that (M, g) is a com- plete Riemannian manifold, the following result is due to de Rham (theorem 24 in [21]) Lemma 1.11.Letα∈Z 2 k (M)∩C ∞ (Λ k T ∗ M)and suppose thatαis zero inH 2 k (M) that is there is a sequenceβ l ∈C 0 ∞ (Λ k−1 T ∗ M)such that α=L 2 −limdβ l l→∞

L 2  9HARMONICS FORMS ON NON COMPACT MANIFOLDS. then there isβ∈C ∞ (Λ k−1 T ∗ M)such that α=dβ. In full generality, we know nothing about the behavior ofβat infinity. Proof.We can always assume thatMytilaeitnsironeecdeh,ePoibyth´eduncar (1.1.2), we only need to show that ifψ∈C 0 ∞ (Λ n−k T ∗ M) is closed then Z M α∧ψ= 0. But by assumption, Z M α∧ψ= l l → im ∞ Z M d β l ∧ψ = lim Z M d(β l ∧ψ) becauseψis closed l→∞ 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  H 0 k (M)→H k (M)  →H 2 k (M). Proof.As a matter of fact we need to show that ifα∈C 0 ∞ (Λ k T ∗ M) is closed and zero in the reducedL 2  is thiscohomology then it is zero in usual cohomology: exactly the statement of the previous lemma (1.11). 1.2.2.Consequence for surfaces.These results have some implications for a com- plete Riemannian surface (S, g) : i) If the genus ofSis infinite then the dimension of the space ofL 2 harmonic 1−forms is infinite. ii) If the space ofL 2 harmonics 1−forms is trivial then the genus ofSis zero andSis diffeomorphic to a open set of the sphere. As a matter of fact, a handle ofSis a embeddingf:S 1 ×[−1,1]→Ssuch that if we denoteA=f(S 1 ×[−1,1]) thenS\ Ais connected.