Reduced $L_{q,p}$-Cohomology of Some Twisted Products

Vanishing results for reduced $L_{p,q}$-cohomology are established in the case of twisted products, which are a~generalization of warped products. Only the case $q \leq p$ is considered. This is an extension of some results by Gol'dshtein, Kuz'minov and Shvedov about the $L_{p}$-cohomology of warped cylinders. One of the main observations is the vanishing of the"middle-dimensional"cohomology for a large class of manifolds. This means that the $L_2$-Betty numbers are zero in the"middle dimension".


Introduction
The L q,p -cohomology H k q,p (M ) of a Riemannian manifold (M, g) is defined to be the quotient of the space of closed p-integrable differential forms by the exterior differentials of q-integrable forms. The quotient space of H k q,p (M ) by the closure of zero is called the reduced L q,p -cohomology H k q,p (M ). In [9], Gol ′ dshtein and Troyanov established some nonvanishing results for the reduced L q,p -cohomology H k q,p (M ) of M in the case of simply-connected complete manifolds with negative curvature. They obtained sufficient conditions for the nonvanishing of the space H k q,p (M ) when M is a Cartan-Hadamard manifold (that is, a complete simply-connected Riemannian manifold of nonpositive sectional curvature). Every Cartan-Hadamard manifold is bi-Lipschitz equivalent to a so-called twisted product outside any geodesic ball.
A twisted product X × h Y of two Riemannian manifolds (X, g X ) and (Y, g Y ) is the direct product manifold X × g Y endowed with a Riemannian metric of the form where h : X × Y → R is a smooth positive function (see [3]). If X is a half-interval [a, b[ then the twisted product X × h Y is called a twisted cylinder.
We refer to an m-dimensional Riemannian manifold (M, g M ) as an asymptotic twisted product (respectively, as an asymptotic twisted cylinder) if, outside an mdimensional compact submanifold, it is bi-Lipschitz equivalent to a twisted product (respectively, to a twisted cylinder). A Cartan-Hadamard manifold is an asymptotic twisted cylinder of this type. Other examples are manifolds with flat ends, manifolds with cuspidal singularities etc. In this paper, we prove some vanishing results for the (reduced) L q,p -cohomology of twisted cylinders [a, b) × h N for a positive smooth function h : [a, b) × N → R in the case where the base N is a closed manifold and p ≥ q > 1.
If in (1.1) the function h depends only on x then we obtain the familiar notion of a warped product (see [1]). Twisted products were the object of recent investigations [2,4,5,6,11,15]. The reduced L q,p -cohomology of warped cylinders [a, b) × h N , i.e., of product manifolds [a, b) × N endowed with a warped product metric where g N is the Riemannian metric of N and h : [a, b) → R is a positive smooth function, was studied by Gol ′ dshtein, Kuz ′ minov, and Shvedov [7], Kuz ′ minov and Shvedov [13,14] (for p = q), and Kopylov [12] for The main results of this paper are technical. Here we mention a "universal" consequence of the main results on the vanishing of the "middle-dimensional" cohomology: Let N be a closed smooth n-dimensional Riemannian manifold. If p ≥ q > 1, b = ∞, and n p is an integer, then H The result was not known even for L 2 -cohomology. It does not depend on the type of the warped Riemannian metric. Of course, the result leads to the vanishing of the "middle-dimensional" cohomology for asymptotic twisted cylinders.

Basic Definitions
In this section, we recall the main definitions and notations.
In what follows, we tacitly assume all manifolds to be oriented. Let M be a smooth Riemannian manifold. Denote by D k (M ) := C ∞ 0 (M, Λ k ) the space of all smooth differential k-forms with compact support contained in M \ ∂M and designate as L 1 loc (M, Λ k ) the space of locally integrable differential forms. Denote by L p (M, Λ k ) the Banach space of locally integrable differential k-forms endowed with the norm θ L p (M,Λ k ) := M |θ| p dx 1 p < ∞ (as usual, we identify forms coinciding outside a set of measure zero).
Remark 2.2. Note that the orientability of M is not substantial since one may take integrals over orientable domains on M instead of integrals over M . We then introduce an analog of Sobolev spaces for differential k-forms, the space of q-integrable forms with p-integrable weak exterior derivative: , This is a Banach space for the graph norm The space Ω k q,p (M ) is a reflexive Banach space for any 1 < q, p < ∞. This can be proved using standard arguments of functional analysis.
Denote by Ω k q,p,0 (M ) the closure of D k (M ) in the norm of Ω k q,p (M ). We now define our basic ingredients (for three parameters r, q, p).
Lemma 2.4. The subspace Z k p,r (M ) does not depend on r and is a closed subspace in L p (M, Λ k ).
Proof. The lemma is in fact [10, Lemma 2.4 (i)]. However, we now repeat the proof for the reader's convenience. Note that Z k p,r (M ) is a closed subspace in Ω k p,r (M ) because it is the kernel of the bounded operator d. It is also a closed subspace of This allows us to use the notation Z k p (M ) for all Z k p,r (M ). Note that Z k p (M ) ⊂ L p (M, Λ k ) is always a closed subspace but this is in general not true for B k q,p (M ). Denote by B k q,p (M ) its closure in the L p -topology. Observe also that since d , and the reduced L q,p -cohomology of (M, g) is, by definition, the space Since B k p,q is not always closed, the L q,p -cohomology is in general a (non-Hausdorff) semi-normed space, while the reduced L q,p -cohomology is a Banach space. Considering only the forms equal to zero on some neighborhood (depending on the form) of a subset A ⊂ M and taking closures in the corresponding spaces, we obtain the definition of the relative spaces L p (M, A, Λ k ) and Ω q,p (M, A) and the relative nonreduced and reduced cohomology spaces H k q,p (M, A) and H k q,p (M, A). Similarly, one can define the L q,p -cohomology with compact support (interior cohomology) H k q,p;0 (M, A) and H k q,p;0 (M, A). The interior reduced cohomology is dual to the reduced cohomology: [10]. Below |X| stands for the volume of a Riemannian manifold (X, g).

L q,p -Cohomology and Smooth Forms
It follows from the results of [8] that, under suitable assumptions on p, q, the L q,p -cohomology of a Riemannian manifold can be expressed in terms of smooth forms.
Introduce the notations: In what follows, unless otherwise specified, we always assume that p, q ∈ (1, ∞)

The Homotopy Operator
From now on, C h a,b N is the twisted cylinder I × h N , that is, the product of a half-interval I := [a, b) and a closed smooth n-dimensional Riemannian manifold (N, g N ) equipped with the Riemannian metric dt 2 +h 2 (t, x)g N , where h : I ×N → R is a smooth positive function.
Every differential form on I × N admits a unique representation of the form ω = ω A + dt ∧ ω B , where the forms ω A and ω B do not contain dt (cf. [7]). It means that ω A and ω B can be viewed as one-parameter families ω A (t) and ω B (t), t ∈ I, of differential forms on N . Given a form ω defined on I × N and numbers c, t ∈ [a, b), consider the form The modulus of a form ω of degree k on C h a,b N is expressed via the moduli of ω A (t) and ω B (t) on N as follows: Consequently, In the particular case of ω = ω A , call the form ω horizontal. If ω is a horizontal form then Remark 4.1.
(2) For warped products (h depends only on x), f k,p (t) = F k,p (t) = h Every smooth k-form ω on C h a,b N satisfies the homotopy relations Here d N stands for the exterior derivative on N and d designates the exterior derivative on [a, b) × N .
The homotopy relations cannot be used automatically for ω ∈ Ω k q,p (C h a,b N ) because of the problem of the existence of traces on submanifolds. However, by Theorem 3.1, we can take only smooth forms in all considerations concerning both reduced and nonreduced L q,p -cohomology.
For the reader's convenience, we repeat the classical proofs of (4.4) and (4.5).
Remark 4.4. The proof shows that Lemma 4.3 actually holds in the case where N is not necessarily compact but has finite volume.

5.1.
Absolute reduced L q,p -cohomology. Using the results of the previous section, we prove Theorem 5.1. Let N be a closed smooth n-dimensional Riemannian manifold and let p ≥ q > 1. If The condition "J δ0,b = ∞ for some δ 0 ∈ (a, b)" is in fact equivalent to "J δ0,b = ∞ for every δ 0 ∈ (a, b)". In this connection, below we sometimes write Proof. Suppose that ω ∈ L k p (C h a,b N ) is weakly differentiable and dω = 0. Therefore, ω ∈ Ω k p,p (C h a,b N ). By Theorem 3.1, we may assume that ω is a smooth form. If Consider the form . It is easy to verify that ω j belongs to Ω k q,p (C h a,b N ). Since dω = 0 and dπi * τj ω = 0, we have dω j = 0.
We infer a,b N ) and dS τj (ω − ω j ) = ω − ω j . Thus, the cocycle ω is zero in the reduced cohomology H Proof. In this case, I a,∞ = J a,∞ = ∞, and by Theorem 5.1 H n p q,p (C h a,∞ N ) = 0. Theorem 5.5. Let N be a closed smooth n-dimensional Riemannian manifold. Assume that p ≥ q > 1,Ĩ Proof. Suppose that ω ∈ C ∞ Ω k p,p (C h a,b N, N a ) and dω = 0. The form ω is the limit in Ω k p,p (C h a,b N ) of a sequence ω j of smooth forms each of which is equal to zero on some neighborhood of N a .

and the integral
As above, for a ≤ c < d ≤ b denote by C h c,d N the product [c, d) × N with the metric induced from C h a,b N . For every e ∈ (a, b), Lemma 4.3 implies that the operator S a : Ω k p, N ). Each of the forms S a ω j vanishes on some neighborhood of N a . Therefore, S a ω ∈ Ω k−1 q,p (C h a,e N, N a ) for all e ∈ (a, b). Then the fact that i * a = 0 and relation (4.5) give the equality dS a ω = ω.
Consider the functions The function ϕ(t) is defined for τ sufficiently close to b and ϕ δ (t) also exists only for δ that are sufficiently close to b.
Since d(ϕ δ S a ω) = dϕ δ ∧ S a ω + ϕ δ ω, it follows that . By (4.2) and Lemma 4.3 for p = q, for δ sufficiently close to b we infer By hypothesis, the last quantity vanishes as δ → b. Thus, d(ϕ δ S a ω) → ω as δ → b. The function ϕ δ is equal to zero in some neighborhood of b. Therefore, ϕ δ S a ω ∈ Ω k q,p (C h a,b N, N a ). This shows that H k q,p (C h a,b N, N a ) = 0. Remark 5.6. The paper [7] contains the following assertion for a warped cylinder C h a,b N [7, Theorem 2]: Thus, Theorem 5.5 generalizes this assertion to twisted cylinders C f a,b N with N closed.

Asymptotic Twisted Cylinders
Definition 6.1. We refer to a pair (M, X) consisting of an m-dimensional manifold M and an n-dimensional compact submanifold X with boundary as an asymptotic twisted cylinder AC h a,b ∂X if M \ X is bi-Lipschitz diffeomorphic equivalent to the twisted cylinder C h a,b ∂X. Theorem 5.5 readily implies Theorem 6.2. Let (M, X) = AC h a,b ∂X be an asymptotic twisted cylinder. Assume

and the integral
Then H k q,p (M, X) = 0. Proof. Note that bi-Lipschitz diffeomorphisms preserve L p and L q . Moreover, extension by zero gives a topological isomorphism between the relative spaces To obtain a version of this theorem for H k q,p (M ), we will need the exact sequences of a pair for L q,p -cohomology.
Recall that an exact sequence of cochain complexes of vector spaces is called an exact sequence of Banach complexes if A, B, C are Banach complexes and all mappings ϕ k , ψ k are bounded linear operators. A short exact sequence (6.1) of Banach complexes induces an exact sequence in cohomology with all operators ∂, ϕ * , ψ * bounded and a sequence in reduced cohomology where all operators ∂, ϕ * , ψ * are bounded and the composition of any two consecutive morphisms is equal to zero. Sequence (6.2) is in general not exact but exactness at its particular terms can be guaranteed by some additional assumptions. For example, we have (see [7, Theorem 1 (1)]): Proposition 6.4. Given an exact sequence (6.1) of Banach complexes, if H k (C) 0 = 0 and dim ∂ k−1 (H k−1 (C)) < ∞ then the sequence is exact.
Let (M, X) = AC h a,b ∂X be an asymptotic twisted cylinder. Consider the short exact sequence of Banach complexes In all the three complexes of (6.3), i is the natural embedding, and j is the restriction of forms. The cohomology of C ∞ Ω P (X) is simply the de Rham cohomology of X.