L p -cohomology and pinching

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1 L -cohomology and inching Pierre Pansu Université de Paris-Sud, UMR 8628 du CNRS, Laboratoire de Mathématiques, Equie de Toologie et Dynamique, Bâtiment 425, F Orsay Cedex, France Abstract. This aer is an exosition of some material from [P]. We exlain how torsion in L -cohomology can be used to rove a shar inching theorem for simly connected Riemannian manifolds with negative curvature. Namely, it is shown that a certain Riemannian homogeneous sace whose curvature is negative and inched cannot be quasiisometric to any Riemannian manifold whose curvature is less than 1 4 -inched. 1 Negative inching Let 1 δ < 0. Say a Riemannian manifold is δ-inched if its sectional curvature lies between a and δa for some a > Examles : Rank one symmetric saces of non comact tye Lobatchevski sace HR n is 1-inched. Comlex hyerbolic sace HC n, n 2, quaternionic hyerbolic sace Hn H, n 2, and Cayley hyerbolic lane HO 2 are 1 4 -inched. 1.2 The motivating roblem On a given manifold, what is the best ossible inching? For simly connected (and thus non comact) manifolds, one must be more secific, and require that the unknown metric g be equivalent in the following sense to some reference metric g 0 : there exists C > 0 such that 1 C g g 0 C. We are fascinated by the following roblem. Problem : On HK n (K R, n 2), does there exist a metric which is δ-inched for δ < 1 4 and equivalent to the symmetric metric? If one assumes the unknown metric to be eriodic (i.e. admit a cocomact isometry grou), then the answer is no, due to M. Ville [V] (dimension 4) and L. Hernández [Hz] (other cases). The general roblem is oen. Note that the corresonding roblem for symmetric saces of comact tye has been solved by M. Berger and W. Klingenberg in 1958, [Be].

2 2 Pierre Pansu 2 Statement of the result We shall describe a related result, where symmetric saces are relaced by a certain class of non-symmetric homogeneous saces. For simlicity, we stick to one examle in dimension 4 (see [P] for more general statements). This examle can be viewed as a Riemannian metric g 1 on R 4, which is both a deformation of the symmetric metric g 0 of comlex hyerbolic lane H 2 C and of the curvature -1 metric g 2 of real hyerbolic (Lobatchevsky) 4-sace H 4 R. 2.1 HC 2 as a left invariant metric on a 4-dimensional solvable Lie grou. Fix a oint ξ on the ideal boundary of HC 2. The subgrou of isometries of HC 2 fixing ξ contains a 3-dimensional nilotent non abelian Lie grou Heis which acts simly transitively on each horoshere with center ξ. Consider a 1-arameter grou R of translations along a geodesic assing through ξ. The roduct G 0 = R Heis Isom(HC 2 ) acts simly transitively on H2 C, and inherits a left-invariant metric g 0 isometric to HC 2. In order to get coordinates, observe that G 0 is a semi-direct roduct of R and Heis. Let t denote some isomorhic R R and write a tyical element of Heis as a uniotent 3 3 matrix 1 x z 0 1 y. Then (t, x, y, z) R 4 form coordinates on G 0, in which the metric g 0 writes g 0 = dt 2 + e 2t (dx 2 + dy 2 ) + e 4t (dz xdy) A quasi comlex-hyerbolic metric For each ε [0, 1[, g ε = dt 2 + e 2t (dx 2 + dy 2 ) + e 4t (dz (1 ε)xdy) 2 is a left-invariant metric on G 0 too, but in different coordinates. Let g 1 = dt 2 + e 2t (dx 2 + dy 2 ) + e 4t dz 2. g 1 is a left-invariant metric on the semi-direct roduct G 1 = R α R 3 where α is the derivation of R 3 with matrix Since G 1 is not isomorhic to G 0, g 1 is not isometric to any left-invariant metric on G 0 although it is a limit of such metrics.

3 L -cohomology and inching Pinched deformations One can also view g 1 as a deformation, among 1 4-inched homogeneous metrics, of real hyerbolic sace g 2, by setting, for ε [1, 2], g ε = dt 2 + e 2t (dx 2 + dy 2 ) + e (6 2ε)t dz 2. Indeed, g ε is a left invariant metric on the semi-direct roduct G ε = R αε R 3 where α ε = A simle comutation (see [He]) shows that the ε sectional curvature of g ε varies between (3 ε) 2 and 1, and is therefore 1 4 -inched for all ε [1, 2]. In other words, the Lie grou G 1 is a continuous deformation of real hyerbolic sace which is infinitely close to comlex hyerbolic sace, u to quasiisometry. 2.4 The shar inching result Theorem 1. No metric on R 4 equivalent to g 1 is δ-inched for some δ < Scheme of roof The roof uses a numerical invariant deduced from the torsion in L -cohomology in degree 2. This is a vector sace T 2, (M, g) attached to a Riemannian manifold (M, g) (see below for a recise definition). It has the following roerties. Fact 1 : If g and g are equivalent metrics on M, then T 2, (M, g) and T 2, (M, g ) are isomorhic. Definition 1. Let τ(m, g) = inf{ > 1; T 2, (M, g) 0}. Fact 2 : If g is δ-inched, then τ(m, g) δ. Theorem 1 immediately follows from Fact 1, Fact 2 and Fact 3 : τ(r 4, g 1 ) = 2. The reason our roof fails for H 2 C is Fact 4 : τ(h 2 C ) = τ(r4, g 0 ) = 4. In the sequel of the talk, we briefly discuss L -cohomology and exlain the four facts about it.

4 4 Pierre Pansu 4 L -cohomology This is the cohomology of the de Rham exterior differential acting on differential forms with a decay condition. It turns out to be significant even if the underlying manifold is contractible. 4.1 Notation Let Ω k, = {k forms ω on M with ω L, dω L }, H k, = (Ω k, Kerd)/dΩ k 1,, R k, = (Ω k, Kerd)/dΩ k 1,, T k, = dω k 1, /dω k 1,. R k,, the reduced cohomology, is a Banach sace. The torsion T k, does not have a norm. It is not a Hausdorff toological sace. 4.2 Examle: The real hyerbolic lane H 2 R Here H 0, = 0 = H 2, for all. Let us begin with = 2. Since the Lalacian on L 2 functions is bounded below, T 1,2 = 0. Therefore H 1,2 = R 1,2 = {L 2 harmonic 1-forms} = {harmonic functions h on H 2 R with L 2 }/R. Since the Dirichlet integral h 2 in 2 dimensions is a conformal invariant, one can switch from the hyerbolic metric on the disk D to the euclidean metric on the disk. Therefore H 1,2 = {harmonic functions h on D with h L 2 }/R = {Fourier series Σa n e inθ with a 0 = 0, Σ n a n 2 < + } which is the Sobolev sace H 1/2 (R/2πZ). More generally, for > 1, T 1, = 0 and H 1, is equal to the Besov sace B, 1/ (R/2πZ) mod constants. 4.3 Examle: the real line In that case, H 0, = 0. R 1, = 0 since every function in L (R) can be aroximated in L with derivatives of comactly suorted functions. Therefore H 1, is only torsion. It is non zero and thus infinite dimensional. Indeed, the 1-form dt t (cut off near the origin) is in L for all > 1 but it is not the differential of a function in L.

5 L -cohomology and inching Riemannian homogeneous saces It seems to be a mixture of the two above behaviours. Torsion is frequent, reduced cohomology seems to be connected with negative curvature, see [Bo], [P]. 4.5 More general saces Quite a number of results and examles exist concerning L 2 -cohomology, see [D], [CG], [G1]. L -cohomology in degree 1 is related to dimension at infinity of saces, see [G2]. In higher degrees, existing results are confined to values of close to 2, see [CL]. 5 The Künneth formula This is our tool for L cohomology calculations. Remember that we view our reference saces (R 4, g 0 ) = H 2 C, (R4, g 1 ), (R 4, g 2 ) = H 4 R as semi direct roducts of the form R α H where H is a 3-dimensional Lie grou and α a derivation of Lie (H). Each of the reference metrics writes g ε or g ε = dt 2 + ex(tα) ds 2 where ex(tα) is the 1-arameter grou of automorhisms of H generated by α. More generally, let M be a 1-connected comlete negatively curved Riemannian manifold. Fix a oint on the ideal boundary of M. Let β denote the corresonding Busemann function. Its gradient β is a unit vector field, with flow ϕ t. Use it to define a global diffeomorhism R H M, (t, h) ϕ t (h) where H = β 1 (0) is a horoshere. In these coordinates (called horosherical coordinates), the metric of M takes the form dt 2 + g t. Thus a 1-connected negatively curved manifold is a roduct sace, equied with a nearly roduct metric. The difficulty is that g t deends on t ; as a consequence, H, (M) H, (R) H, (H) in general. 5.1 Examle: Real hyerbolic sace We exlain the comutation of cohomology on an examle, n-dimensional real hyerbolic sace. In horosherical coordinates, g = g 2 = dt 2 + e 2t (dx dx 2 n 1). In other words, the flow ϕ t is isometric in the direction of t = β and dilates by a factor e t orthogonally. In articular, the metric in far from a Riemannian roduct.

6 6 Pierre Pansu Inverting the exterior differential. The Künneth formula in homological algebra suggests that, in order to solve the exterior differential d = dt ± dn 1 t where d n 1 denotes the exterior differential on forms on the R n 1 -factor, one should first invert the differential on the first factor, i.e. t = L β (Lie derivative). The solution θ of the equation L β θ = γ takes the form θ = ϕ t i β γ dt. If β is a k-form in L, i β γ is a (k 1)-form with no comonent along β, therefore ϕ t i β γ L = e ( n 1 (k 1))t i β γ L. Thus, if n 1 If n 1 (k 1) < 0, one can ut θ = (k 1) > 0, one uts + 0 ϕ t i β γdt =: Bγ. 0 θ = ϕ t i β γdt =: Bγ. In both cases, B is a bounded oerator on L differential k-forms. The Besov comlex. Let P = 1 db Bd. If n 1 (k 1) 0 for all k = 1,..., n, this is a homotoy equivalence of the comlex (Ω k,, d) to the comlex (B, d) of differential forms ω on HR n such that 1. L β ω = 0, L β dω = 0; 2. ω has -1 derivative in L, i.e. 1/2 ω L. The loss of differentiability cannot be avoided, since db is not bounded on L. We call B, the Besov comlex. Calculating the Besov comlex. Condition (1) means that ω is ulled back from the hyersurface H = R n 1 under the rojection H n R H. Condition (2) can be written: ω = γ + dε where γ, ε L, γ is a k-from and ε is a (k 1)-form. This imlies that for all t, ω = ϕ t ω = ϕ t γ + dϕ t ε, and thus ω L ϕ 1 t γ L + ϕ t ε L e ( n 1 k)t γ L + e ( n 1 (k 1))t ε L.

7 L -cohomology and inching 7 If n 1 k and n 1 (k 1) have the same sign, then the exonentials can be made simultaneously small, and ω = 0. This shows that [ n 1 n 1 (k 1) < 0 or n 1 k 0 B k, = 0. Therefore, the comlex B, has only one nonzero grou, in degree k = 1]. In fact,, n 1 Theorem 2. Let k = 1,..., n. Then H k, (HR n n 1 ) = 0 unless ] k k 1 ]. In = n 1 k 1, Hk, (HR n n 1 ) is all torsion. If ] k, n 1 k 1 [, Hk, (HR n ) is reduced. It is the sace of closed k-forms on the shere S n 1 with coefficients in the Besov sace B n 1 k+,. 5.2 (k, )-Anosov flows The receding construction can be generalized to saces equied with a flow which exonentially contracts or dilates transverse differential forms. In fact, one can deal with cases where certain forms are contracted and others dilated. This is crucial to handle comlex hyerbolic lane, for instance. Definition 2. Let ξ be a unit vector field on a Riemannian manifold M, let k = 0,..., n = dim M, let > 1. Say ξ is (k, )-Anosov if orthogonally to ξ, k-forms slit as Λ k + Λ k where forms in Λ k + are contracted (in L -norm) and form in Λ k are dilated by the flow ϕ t of ξ, i.e. C, η > 0 such that for ω Λ k +, ϕ t ω L Ce ηt ω L for all t 0 ; for ω Λ k, ϕ t ω L Ce ηt ω L for all t Examles: Model saces (R 4, g 0 ) and (R 4, g 1 ) In those examles, the Busemann vectorfield β = / t is (1, ) Anosov for all 2 and 4. The subbundle Λ 1 + is everything if > 4, is zero if < 2, and is generated by dz xdy (res. by dz) if 2 < < 4. Note that no difference between the models is visible yet, as far as exonents are concerned. What will make a difference in torsion is the fact that dz is closed whereas dz xdy is not. 5.4 The Künneth formula Essentially, it says that given a (, )-Anosov vector field on M, the L - cohomology of M is equal to the cohomology of the Besov comlex B,. The roof, insired by V. Livšič solution of the cohomology roblem for Anosov flows, [Li], amounts to generalizing the B oerator described in the

8 8 Pierre Pansu secial case of constant curvature: we integrate searately the Λ + and Λ comonents of a form. The Besov comlex itself is not easy to comute. We include in the following theorem some artial information. Recall that the Besov comex can be viewed as a function sace of differential forms on some hyersurface. On the hyersurface, k-forms slit as Ω k = Ω k + Ω k (where Ω k + is the sace of sections of Λ k +). Let d + denote exterior differential followed by rojection to Ω +. Theorem 3. Let M be a comlete Riemannian manifold, ξ a unit vector field on M. Assume ξ is (k, )-Anosov for all k n 1. Then the L comlex Ω, on M is homotoy equivalent to the Besov comlex B,. Furthermore, k n 1, B, d + Ω k 1 d Ω k Pinched manifolds Let M n be a δ-inched comlete simly connected Riemannian manifold. Let ξ be the Busemann vector field relative to some oint of the ideal boundary of M. From the Rauch comarison theorem, it follows that ξ is (k, ) Anosov with Λ k + = 0 if < 1 + n 1 k δ ; k ξ is (k, ) Anosov with Λ k = 0 if > 1 + n 1 k k. δ If < 1 + n k k 1 δ, then Λ k 1 + = 0, thus d + = Ω k 1 Ω k is zero. This imlies that B k 1, = 0. In articular H k, (M) B k, Kerd is Hausdorff, and T k, (M) = 0. This is fact 2 when n = 4, k = 2. The fact that inching imlies vanishing of L 2 -cohomology was first observed by H. Donnelly and F. Xavier, [DX]. However, only the vanishing of torsion gives a shar inching result. 7 Non vanishing of torsion 7.1 Poincaré duality We use the following form of Poincaré duality, due to V. Gol dstein and M. Troyanov, [GT].

9 L -cohomology and inching 9 Lemma 1. Let M n be an oriented comlete Riemannian manifold. Let ω Ω k, (M) be a closed k form in L. Then ω 0 in L cohomology if and only if there exists a sequence ω j (M) (where 1 Ωn k, + 1 = 1) such that M ω ω j does not tend to zero ; dω j L tends to zero. 7.2 Torsion of roducts On a Riemannian roduct sace M = R H, one constructs non trivial cohomology classes as follows. Let χ be a function on R which is 0 near and 1 near + then dχ = χ (t)dt generates H 1 c (R). Let e be a closed (k 1)- form on H which is non zero in H k 1, (H). Let π denote the rojection onto the second factor. Then ω = dχ π e is a closed k-form in L (R H). To show that ω 0 in H k, (R H), let e j be a sequence of (n k+1)-forms on H such that e e j = 0, let χ j be cut off functions on R converging to 1 on comact sets, but such that dχ j L (R) 0. We view them as functions on R H. Let ω j = χ jπ e j. Then ω ω j = 1 for j large enough, dω j = dχ j π e j + χ j π de j dω j L dχ j L (R) e j L (H) + χ j L (R) de j L (H). We can conclude with Lemma 1 rovided e j and χ j do not tend to infinity too fast, which can often be achieved. 7.3 Semi direct roducts We consider semi-roducts of Lie grous of the form R α H. Since the metric along H diverges exonentially, the fact that H 1, (R) is torsion cannot be used any more. Instead, one exloits exonential decay. One relaces π e j by a form φ j which is in Λ + where t is close to +, and in Λ where t is close to, as follows. Let f j be a (k 2)-form on H, and let φ j = χπ d + f j + (1 χ)π d f j + dχ π f j. Then let ω j = χ jφ j where dχ j is merely required to have its suort near the boundary of a large interval. This leads to Lemma 2. Let G = R α H be a semi-direct roduct of Lie grous, with H nilotent. Let e Ω k 1, (H) be a closed (k 1)-form in L (H). Assume their exists a sequence f j Ωn k 1, (H) of differential (k 2)-forms such that 1. H e d +f j = 1

10 10 Pierre Pansu 2. d + f j L (H) tends to + at most olynomially in j 3. dd + f j L (H) tends to 0 exonentially. Then ω = dχ π e gives a non vanishing class in T k, (G). 7.4 Torsion for the homogeneous sace (R 4, g 1 ) According to Poincaré duality, fact 3 is equivalent to Proosition 1. If 4 3 < < 2, T 3, (R 4, g 1 ) 0. Proof of fact 3: Here, H = R 3 is euclidean sace with coordinates x, y, z. Let e = d(fhdy) where f is a function of the distance to the origin, h a function which is homogeneous of degree 0. The functions f j deend only on the distance to the origin and are carefully chosen in order that the assumtions of lemma 2 be satisfied. 8 Vanishing of torsion for H 2 C The receding construction does not extend to HC 2 = (R4, g 0 ). Indeed, in that case, H is the Heisenberg grou. Let X, Y, Z = [X, Y ] be a basis of left-invariant vector fields on H, and dx, dy, τ = dz xdy be the dual basis of invariant 1-forms. Let f j be a function on H. Then df j = (Xf j)dx + (Y f j)dy + (Zf j)τ d + f j = (Zf j)τ dd + f j = d(zf j) τ + (Zf j)dτ. In articular, Zf j = dd +f j (X Y ). Thus d + f j L dd +f j L and R 3 e d + f j must tend to 0 if dd +f j L does. The same mechanism imlies that torsion vanishes. Proosition 2. Let 2 < < 4. Then T 2, (R 4, g 0 ) = 0. Proof : Since Λ 0 + = 0 and Λ 1 + is sanned by τ, Ω+ 0 = 0, d Ω+ 0 = 0, B 1, d + Ω 0 Ω+ 1. A tyical element of B 1, can be written e = fτ where f is a distribution on the Heisenberg grou. Then de = df τ + fdτ, f = de(x Y ). We show that there exist ositive constants c 1 and c 2 such that c 1 π e L 1 f B c 2 π de L 1.

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