The spectrum of the Laplacian on forms over flat manifolds
aa r X i v : . [ m a t h . DG ] O c t THE SPECTRUM OF THE LAPLACIAN ON FORMS OVER FLATMANIFOLDS
NELIA CHARALAMBOUS AND ZHIQIN LU
Abstract.
In this article we prove that the spectrum of the Laplacian on k -formsover a noncompact flat manifold is always a connected closed interval of the non-negative real line. The proof is based on a detailed decomposition of the structureof flat manifolds. Introduction
We consider the spectrum of the Hodge Laplacian ∆ on differential forms of anyorder k over a noncompact complete flat manifold M . It is well known that theLaplacian is a densely defined, self-adjoint and nonnegative operator on the space of L integrable k -forms. The spectrum of the Laplacian consists of all points λ ∈ R for which ∆ − λI fails to be invertible. The essential spectrum consists of the clusterpoints in the spectrum and of isolated eigenvalues of infinite multiplicity. We will bedenoting the spectrum of the Laplacian on k -forms over M by σ ( k, M ) and its essentialspectrum by σ ess ( k, M ). The complement of the essential spectrum in σ ( k, M ), whichconsists of isolated eigenvalues of finite multiplicity, is called the pure point spectrumand is denoted by σ pt ( k, M ). Since ∆ is nonnegative, its spectrum is contained inthe nonnegative real line. The spectrum, the essential spectrum, and the pure pointspectrum are closed subsets of R .When the manifold is compact, the essential spectrum is an empty set and thespectrum consists only of discrete eigenvalues. In the case of a noncompact completemanifold, on the other hand, a continuous part in the spectrum might appear. Unlikethe point spectrum, which in most cases cannot be explicitly computed, the essential Date : April 15, 2019.2010
Mathematics Subject Classification.
Primary: 58J50; Secondary: 53C35.
Key words and phrases. essential spectrum, Hodge Laplacian, flat manifolds.The first author was partially supported by a University of Cyprus Start-Up grant. The secondauthor is partially supported by the DMS-1510232. spectrum can be located either by the classical Weyl criterion as in [4], or by ageneralization of it as we have shown in previous work [1]. Both criteria requirethe construction of a large class of test differential forms that act as generalizedeigenforms.Our main goal in this article is to compute the spectrum and essential spectrum ofa general noncompact complete flat manifold M n . The main result of this paper isthe following: Theorem 1.1.
Let M be a flat noncompact complete Riemannian manifold. Then σ ( k, M ) = σ ess ( k, M ) = [ α, ∞ ) for some nonnegative constant α . The constant α in the above theorem is the first eigenvalue on ℓ -forms ( ℓ ≤ k ) forsome compact flat manifold that reflects the structure of M at infinity. In Section 4we will give a more precise description of it.The structure of a complete flat manifold may be understood through the results ofCheeger-Gromoll, by which flat manifolds are diffeomorphic to normal bundles overtotally geodesic compact submanifolds known as the soul. It is also well known thata complete flat manifold is given as the quotient of Euclidean space R n by a discretesubgroup Γ of the Euclidean group E ( n ). The properties of Γ, when it is cocompact,are described by the Bieberbach Theorems (see [7, 18]). We recall that in order for thequotient to be a complete manifold, Γ must be discrete and fixed point free. If M canbe decomposed as the direct product of a compact manifold Z n − s times a Euclideanspace R s , then the spectrum of the Laplacian on forms can be computed as we showin Lemma 2.1. In this case, the pure point spectrum is empty and the spectrum ofthe Laplacian on k -forms is a closed connected interval as in Theorem 1.1.A general flat manifold however, cannot always be decomposed in product form.Moreover, the classification of non-compact flat manifolds is far from being complete.Thurston illustrates that the soul of the manifold is essentially the quotient of Eu-clidean space modulo an abelian subgroup of Γ, and proves that the manifold is anormal flat bundle over this compact flat submanifold [17]. Mazzeo and Phillips usethis decomposition when computing the spectrum of the Laplacian on forms overhyperbolic manifolds in [14]. However, the normal bundle structure of a general flatmanifold may be quite complicated, and the soul of the manifold might not provideus with neither sufficient nor useful information when computing the spectrum of theLaplacian on forms (see Example 2.2). HE SPECTRUM OF THE LAPLACIAN ON FORMS OVER FLAT MANIFOLDS 3
One of the main accomplishments of this article is to find an adequate subgroupΓ of the group Γ, so that the spectrum of the flat manifold R n / Γ coincides withthat of R n / Γ . Our choice of Γ will be such that R n / Γ is a product manifold whosespectrum and essential spectrum can be computed. Finding Γ is the crux of ourresults. Although we begin our analysis of Γ by using the subgroup Γ ∗ introducedin J. Wolf [18], (the same group is also used by Thurston and Mazzeo-Phillips todescribe the soul of the manifold [14, 17]) we need to take our decomposition one stepfurther and a more detailed analysis is necessary. The details of this construction canbe found in Section 3.Over a noncompact manifold the essential spectrum of the Laplacian on functionscan present gaps [11, 12, 15, 16]. These manifolds do not have nonnegative Riccicurvature, but some of them have bounded positive scalar curvature. On the otherhand, it is not uncommon for the spectrum of the Laplacian on forms to have gaps.In Example 2.3, over certain even dimensional quotients of hyperbolic space H N +1 ,for the half dimension k = ( N + 1) /
2, the essential spectrum of the Laplacian on k -forms is not a connected set. Given all the above known examples, the result of ourpaper is somewhat surprising. We show that the spectrum will always be a connectedinterval for a noncompact complete flat manifold. In a forthcoming paper we willillustrate the significance of this result as it will become essential for studying thespectrum of the Laplacian on forms over asymptotically flat manifolds. The latter isa significantly more difficult task as can be seen by comparing the difference betweencompact flat manifolds and compact almost flat manifolds. Acknowledgement.
The authors would like to thank V. Kapovich and R. Mazzeofor their feedback and useful discussion regarding the structure of flat manifolds. Theyare also grateful to J. Lott for helping them work out Example 2.4.2.
The Spectrum of a Product Flat Manifold
Let M n = Z n − s × R s be the product of a compact manifold Z n − s of dimension n − s and Euclidean space R s of dimnension s for some 1 ≤ s < n , endowed withthe product metric. We denote the Laplace operator on Z by ∆ Z and on R s by ∆ R s .Since Z is a compact manifold, the spectrum of ∆ Z on ℓ -forms is discrete and wedenote by λ Z ( ℓ ) the smallest eigenvalue of ∆ Z on ℓ -forms, for 0 ≤ ℓ ≤ n − s . Recallthat λ Z ( ℓ ) = λ Z ( n − s − ℓ ) by Poincar´e duality. Note for example that λ Z (0) = 0, not the first nonzero eigenvalue of the compact manifold Z . NELIA CHARALAMBOUS AND ZHIQIN LU
Define α ( Z, s, n, k ) = 0 when s ≥ n/
2, or when s < n/ ≤ k ≤ s ,or n − s ≤ k ≤ n . Define α ( Z, s, n, k ) = min { λ Z ( ℓ ) | k − s ≤ ℓ ≤ k } , when s + 1 ≤ k ≤ n/
2, and α ( Z, s, n, k ) = α ( Z, s, n, n − k ), when n/ < k ≤ n − s − Lemma 2.1.
Let M = M n = Z n − s × R s as above. Let ≤ k ≤ n . Then σ pt ( k, M ) = ∅ , and σ ( k, M ) = σ ess ( k, M ) = [ α ( Z, s, n, k ) , ∞ ) . Proof.
By Poincar´e duality, we know that σ ( k, M ) = σ ( n − k, M ) . Therefore whenever s ≥ n/
2, or when s < n/ ≤ k ≤ s or n − s ≤ k ≤ n ,we can use the duality to reduce computing the spectrum to the case 0 ≤ k ≤ s . Let x , · · · , x s be the coordinates on R s . Since σ (0 , R s ) = [0 , ∞ ), it is well-known thatfor any λ ∈ R + and any ε >
0, there is a function f = 0 with compact support on R s such that k ∆ R s f − λf k L ≤ ε k f k L . Using the test form ω = f ( x ) dx ∧ · · · ∧ dx k on R s and hence on M = Z × R s , we have k ∆ ω − λω k L ≤ Cε k ω k L , where C > n . It follows that σ ( k, M ) = [0 , ∞ ) . Now we assume s < n/ s + 1 ≤ k ≤ n/
2. Since M = Z × R s , we have∆ M = ∆ Z ⊗ ⊗ ∆ R s . Therefore, σ ( k, M ) = k [ ℓ = k − s (cid:0) σ ( ℓ, Z n − s ) + σ ( k − ℓ, R s ) (cid:1) . Since σ ( k − ℓ, R s ) is always [0 , ∞ ), we have σ ( k, M ) = [ α, ∞ ) , where α = α ( Z, s, n, k ). Finally, the case n/ < k ≤ n − s − A special choice of f ( x ) is given in the proof of Proposition 4.1. HE SPECTRUM OF THE LAPLACIAN ON FORMS OVER FLAT MANIFOLDS 5
The result for the essential spectrum of M follows in a similar manner, because M is translation invariant. Since the essential spectrum coincides with the spectrum, thepure point spectrum is an empty set. The translation invariance of M also impliesthat the point spectrum, the set of eigenvalues of finite multiplicity, is an emptyset. (cid:3) As we have mentioned, for the computation of the essential spectrum of the Lapla-cian we must consider the structure of the flat manifold with further detail. Theexample below illustrates why it is important to not simply consider the manifold asa flat bundle over its soul.
Example 2.2.
Consider M = R / Γ where Γ is generated by a glide rotation that iscomposed of an irrational rotation in the xy -plane combined with a translation alongthe z axis. Note that near infinity this flat manifold is essential isometric to R , sinceits injectivity radius becomes infinite.At the same time, the maximal abelian subgroup Γ ∗ of Γ is the translation group Z and the soul of the manifold is the circle, S . On the other hand, the group Γ thatwe describe in the following section, consists only of the identity element. Therefore,the structure of M at infinity is more accurately described by the quotient R / Γ .One can easily see from this example that if the flat manifold is not a parallelnormal bundle over the soul, then the soul need not determine its spectrum. When the manifold has negative curvature, the spectrum of the Laplacian on formscan have gaps. This occurs even over hyperbolic space as the following exampleillustrates.
Example 2.3.
The essential spectrum of the Laplacian on forms over the hyperbolicspace H N +1 is given by σ e ss ( k, ∆) = σ e ss ( N + 1 − k, ∆) = [ ( N − k ) , ∞ ) for ≤ k ≤ N , and whenever N is odd σ e ss ( N + 12 , ∆) = { } ∪ [ 14 , ∞ ) . This result can be found in Donnelly [6] . Mazzeo and Phillips also show that the sameresult is true over hyperbolic manifolds (in other words quotients of hyperbolic space H N +1 / Γ ) that are geometrically finite and have infinite volume [14] . NELIA CHARALAMBOUS AND ZHIQIN LU
At the same time there was an extensive study of curvature conditions on the man-ifold so that the essential spectrum of the Laplacian on functions is the nonnegativereal line [1, 8, 9, 13]. Escobar and Freire also found sufficient curvature assumptionsso that the essential spectrum of the Laplacian on forms is [0 , ∞ ). The above Lemmagives however a very simple example of a flat manifold for which the spectrum of theLaplacian on 2-forms is not [0 , ∞ ). Example 2.4.
Consider the product manifold M = F × R , where F is the com-pact flat three-manifold constructed by Hantzsche and Wendt in 1935 with first Bettinumber zero (see [2] for a family of manifolds of any dimension n ≥ with the sameproperty). Note that the second Betti number of F is also zero by Poincar´e duality.As a result, the first eigenvalue of the Laplacian on 1-forms, λ F (1) , is strictly posi-tive and in fact λ F (1) = λ F (2) > . The product manifold M is a flat noncompactmanifold. By Lemma 2.1 σ e ss ( k, ∆) = [0 , ∞ ) for k = 0 , , , . However, since there do not exist any harmonic 1-forms nor harmonic 2-forms on F then σ e ss (2 , ∆) = [ λ F (1) , ∞ ) where λ F (1) = λ F (2) > . In other words, its essential spectrum is smaller in half-dimension.
This example illustrates the stronger connection of the spectrum of the Laplacianon forms to the topology of the manifold and shows that sufficient conditions so thatthe form spectrum is [0 , ∞ ) must be stricter than for the case of functions.3. A Characterization of Noncompact Flat Manifolds on Large Sets
We will now consider the general case of complete flat manifolds. Recall thata complete flat manifold is given as the quotient of Euclidean space by a discretesubgroup Γ of the Euclidean group E ( n ). As has already been mentioned, Γ must bediscrete and fixed point free in order for the quotient space to be a complete manifold.Fixing a reference point on the Euclidean space each element of E ( n ) can beuniquely represented by ( g, a ) where g ∈ O ( n ) and a ∈ R n . The action of ( g, a )is given by ( g, a ) x = gx + a for any x ∈ R n . Note that R n and hence any sub-space of R n can be embedded in E ( n ) by a (1 , a ). Define the homomorphism ψ : E ( n ) → O ( n ) by ψ ( g, a ) = g .We define Γ ∗ to be the intersection of Γ with the identity component of the closureof Γ · R n . By [18, Theorem 3.2.8], we know that Γ ∗ is a normal subgroup Γ of finite HE SPECTRUM OF THE LAPLACIAN ON FORMS OVER FLAT MANIFOLDS 7 index and there exists a vector subspace V ⊂ R n and a toral subgroup T of O ( n )such that T acts trivially on V , Γ ∗ ⊂ T · V and Γ ∗ is isomorphic to a discrete uniformsubgroup of V .With V defined as above we get the orthogonal decomposition R n = V ⊥ ⊕ V. Let Γ = { ( g, a ) ∈ Γ | g leaves V ⊥ invariant , i.e. g | V ⊥ = I V ⊥ } and define Γ ∗∗ = Γ ∗ ∩ Γ . Lemma 3.1. Γ is a subgroup of Γ , and Γ ∗∗ is a normal subgroup of Γ of finiteindex. Moreover, Γ ∗∗ is a subgroup of the translation group which acts on V ; in otherwords it acts trivially on V ⊥ .Proof. Let ( g , a ) , ( g , a ) ∈ Γ . Then g g | V ⊥ = I V ⊥ , and hence Γ is a subgroup. By [18, Theorem 3.2.8], since Γ ∗ is normal in Γ, thenΓ ∗∗ in also normal in Γ . Moreover, we have[Γ , Γ ∗∗ ] ≤ [Γ , Γ ∗ ] < ∞ . Finally, by definition, ψ (Γ ∗∗ ) is the identity matrix. Therefore Γ ∗∗ ⊂ Γ ∗ is a transla-tion group acting on V . (cid:3) Define V = span (Γ ∗∗ ) , and let V = V ⊕ V be the orthogonal decomposition of V . In this way, we write(1) R n = V ⊥ ⊕ V ⊕ V . Define the subgroup Γ of Γ byΓ = { ( g, a ) ∈ Γ | g leaves V invariant , i.e. g | V = I V } . For any vector a ∈ R n , we decompose a = a ⊥ + a = = a ⊥ + a + a , NELIA CHARALAMBOUS AND ZHIQIN LU where a ⊥ ∈ V ⊥ , a = ∈ V, a ∈ V and a ∈ V . Let(2) d ⊥ = dim V ⊥ , d = dim V , d = dim V . We also choose a coordinate system(3) a = ( a , · · · , a d ⊥ , a d ⊥ +1 , · · · , a d ⊥ + d , a d ⊥ + d +1 , · · · , a n )so as to be compatible with the decomposition (1). Here for 1 ≤ j ≤ n , a j is the j -thcomponent of a .Let g ∈ O ( n ), we use g ⊥ , g = , g , and g to represent the restrictions of the operatorto V ⊥ , V , V , and V , respectively. Note that since g is orthogonal, if the restric-tion is also an orthogonal matrix, then the corresponding space and its orthogonalcomplement are invariant spaces of g . Lemma 3.2.
Let ( g, a ) ∈ Γ . Then there exists a constant C such that k a ⊥ k ≤ C. Moreover, if ( g, a ) ∈ Γ , then a ⊥ = 0 .Proof. We first consider the case ( g, a ) ∈ Γ , and we write( g, a ) = (cid:18)(cid:18) g = (cid:19) , (cid:18) a ⊥ a = (cid:19)(cid:19) , Since Γ ∗∗ is of finite index in Γ , by Lemma 3.1, we have( g, a ) N = (cid:18)(cid:18) (cid:19) , (cid:18) N a ⊥ ∗ (cid:19)(cid:19) ∈ Γ ∗∗ ⊂ Γ ∗ . for some positive integer N . By Lemma 3.1, Γ ∗∗ is a subgroup of the translationgroup that acts only on V , therefore N a ⊥ = 0; hence a ⊥ = 0.Now we assume that ( g, a ) ∈ Γ. Γ ∗ is normal and of finite index in Γ. Therefore,there exists a finite set { ( g ′ α , a ′ α ) } ⊂ Γ such that( g, a ) = ( h, b )( g ′ α , a ′ α )for some α and ( h, b ) ∈ Γ ∗ . By [18, Theorem 3.2.8], we have a ⊥ = ( ha ′ α ) ⊥ + b ⊥ = ( ha ′ α ) ⊥ . Since the set { a ′ α } is finite and h ∈ O ( n ), we conclude that k a ⊥ k is bounded. (cid:3) By the same argument we have
HE SPECTRUM OF THE LAPLACIAN ON FORMS OVER FLAT MANIFOLDS 9
Lemma 3.3.
Let ( g, a ) ∈ Γ . Then there exists a constant C such that k a k ≤ C. Moreover, if ( g, a ) ∈ Γ , then a = 0 . (cid:3) By the above two lemmas, we have the decomposition R n / Γ = V ⊥ ⊕ V ⊕ V / Γ . For a vector x ∈ V ⊥ ⊕ V we write˜ x = ( x ⊥ , x , ∈ R n . Let ˆ T be the closure ψ (Γ), where ψ is the projection of E ( n ) to O ( n ) defined before.By [18, Theorem 3.2.8], ˆ T is a finite extension of the toral group T . Lemma 3.4.
There exists an x ∈ V ⊥ ⊕ V such that for all g ∈ ˆ T ,(1) If ( g ˜ x − ˜ x ) ⊥ = 0 , then g ⊥ = I V ⊥ ;(2) If both ( g ˜ x − ˜ x ) ⊥ = 0 and ( g ˜ x − ˜ x ) = 0 , then both g ⊥ and g are the identity. The points x described in the above lemma are sufficiently generic in the sense thatone can find them in any open set. The essence of the argument is the observationthat ˆ T is a finite extension of a toral group, and as a result it suffices to only considerthe representatives of ˆ T over T . Even though the elements of T may be uncountable,they are easier to control because they are diagonalizable under a fixed coordinatesystem. Proof.
We let p , · · · , p t ∈ ˆ T be the representatives of the group ˆ T /T . We assumethat(1) p = I ;(2) p i | V ⊥ = I V ⊥ and p i | V = I V for 1 ≤ i ≤ t ;(3) p i | V ⊥ = I V ⊥ but p i | V = I V ; moreover, for all h ∈ T , hp i belongs to neithercategory (1) nor (2) above, for t < i ≤ t ;(4) p i | V ⊥ = I V ⊥ ; moreover, for all h ∈ T , hp i belongs to neither category (1), (2)nor (3) above for t < i ≤ t .Obviously, there exists an x ∈ V ⊥ ⊕ V such that(1) ( p i ˜ x − ˜ x ) ⊥ = 0 for t < i ≤ t ;(2) ( p i ˜ x − ˜ x ) = 0 for t < i ≤ t . By continuity, there exists a neighborhood U of x on which the above two conditionsare still held. Therefore, without loss of generality, we assume that all the componentsof x are not zero.Since T is the toral group, we may assume that under the coordinate system (3)for R n , its elements can be represented by(4) S θ . . . S θ r I where S θ i = (cid:18) cos θ i sin θ i − sin θ i cos θ i (cid:19) , and 2 r ≤ d ⊥ . We write V ⊥ = P ⊕ P ⊕ · · · ⊕ P r ⊕ P r +1 according to the above representation. We prove by contradiction. Let g be anelement of ˆ T such that ( g ˜ x − ˜ x ) ⊥ = 0. We assume that g = hp i for some h ∈ T and i .Assume that i > t . Then there exists an 1 ≤ j ≤ r + 1 such that P j is not invariantunder p i . If j ≤ r , then(( p i ˜ x ) j − ) + (( p i ˜ x ) j ) = ((˜ x ) j − ) + ((˜ x ) j ) . As a result, the projection of ( g ˜ x − ˜ x ) ⊥ = ( hp i ˜ x − ˜ x ) ⊥ to P j is not zero, whichcontradicts the assumption. Similarly, if P r +1 is not invariant under p i , then again( g ˜ x − ˜ x ) ⊥ = 0. This proves the first part of the lemma.Similarly, we can prove the case when both ( g ˜ x − ˜ x ) ⊥ and ( g ˜ x − ˜ x ) are zero. Thelemma is proved. (cid:3) Lemma 3.5.
Let x be as in the above lemma. Then lim j →∞ inf ( g,a ) ∈ Γ \ Γ k ( g, a )( j ˜ x ) − j ˜ x k = ∞ . Proof.
If the lemma is false, then there exists a subsequence { λ j } of positive integers λ j → ∞ , ( g j , a j ) ∈ Γ \ Γ , such that(5) k ( g j , a j )( λ j ˜ x ) − λ j ˜ x k ≤ C. HE SPECTRUM OF THE LAPLACIAN ON FORMS OVER FLAT MANIFOLDS 11
For the same N as in the proof of Lemma 3.1, ( g j , a j ) N = ( h j , b j ) ∈ Γ ∗ . We willconsider two cases.In the first case we assume ( h j , b j ) / ∈ Γ for a subsequence of the ( h j , b j ) for whichwe use the same notation. We have k ( h j , b j )( λ j ˜ x ) − λ j ˜ x k ≤ N C since ( g j , a j ) is an isometry. By orthogonality we get(6) k λ j ( h j ˜ x − ˜ x ) ⊥ + ( b j ) ⊥ k ≤ N C and(7) k λ j ( h j ˜ x − ˜ x ) = + ( b j ) = k ≤ N C.
By Lemma 3.2, the upper bound (6) is equivalent to k λ j ( h j ˜ x − ˜ x ) ⊥ k ≤ N C.
Since λ j → ∞ , we must have(8) k ( h j ˜ x − ˜ x ) ⊥ k → j → ∞ . Given that ( h j , b j ) ∈ Γ ∗ we know ( h j ˜ x ) = = (˜ x ) = . Using (7) we get k ( b j ) = k ≤ N C . Applying Lemma 3.2 once again, we obtain the uniform upper bound k b j k ≤ k ( b j ) ⊥ k + k ( b j ) = k ≤ C for some constant C . Γ ∗ as well as Γ are discrete groups, therefore there exist onlyfinitely many ( h j , b j ) with b j bounded. This implies that for sufficiently large j wehave ( h j ˜ x − ˜ x ) ⊥ = 0, which is only possible, by Lemma 3.4, when we have ( h j ) ⊥ = I V ⊥ .Therefore ( h j , b j ) ∈ Γ and we get a contradiction.We now consider the case ( h j , b j ) ∈ Γ for all sufficiently large j . This implies that( h j , b j ) ∈ Γ ∩ Γ ∗ = Γ ∗∗ . As a result g Nj = h j = I R n . On the other hand, using (5),we get k λ j ( g j ˜ x − ˜ x ) ⊥ k ≤ C for all j , which again imples that(9) k ( g j ˜ x − ˜ x ) ⊥ k → j → ∞ .Since Γ is finite over Γ ∗ , we can write( g j , a j ) = ( g ′ j , a ′ j )( p j , q j ) for ( p j , q j ) ∈ Γ ∗ and finitely many ( g ′ j , a ′ j ) . By passing to a subsequence if necessary,we assume that g j → g ∞ , g ′ j → g ′∞ , p j → p ∞ . By (9) and Lemma 3.4, we have g ∞ | V ⊥ = I V ⊥ . Since g ∞ = g ′∞ p ∞ and V ⊥ is aninvariant space for p ∞ , it is also an invariant subspace for g ′∞ . However, since thereexist only finitely many g ′ j , we must have g ′ j = g ′∞ for j ≫
0. Thus V ⊥ is an invariantspace for g ′ j for j ≫
0. As a result, V ⊥ is an invariant space for g j = g ′ j p j for j ≫ g Nj = I V ⊥ and g j satisfies (9), we conclude that g j = I V ⊥ for j ≫ g ∞ ˜ x − ˜ x ) = 0. Using Lemma 3.4 we get g ∞ | V = I V , and hence g ∞ | V ⊥ ⊕ V = I V ⊥ ⊕ V . Since g Nj = I R n and g j → g ∞ with g ∞ | V ⊥ ⊕ V = I V ⊥ ⊕ V , we must have g j | V ⊥ ⊕ V = I V ⊥ ⊕ V for j ≫
0. Thus ( g j , a j ) ∈ Γ ,which is a contradiction. (cid:3) Theorem 3.6.
Let M = R n / Γ be a flat noncompact Riemannian manifold. Thenthere exists a compact flat manifold Z of dimension n − s , such that for any sufficientlylarge real number R > , there exists an isometric embedding Z × B s ( R ) → R n / Γ , where B s ( R ) is a ball of radius R in the Euclidean space R s .Proof. We take Z = V / Γ which is compact given our choice of Γ . Note that s = d ⊥ + d . Choose any C > Z satisfies diam( Z ) ≤ C .By Lemma 3.5 there exists a λ > ( g,a ) ∈ Γ \ Γ k ( g, a )( λ ˜ x ) − λ ˜ x k ≥ C. This implies that the ball of radius C/ R n / Γ can be isometrically embedded into R n / Γ. (cid:3) Computation of the Spectrum
The isometric embedding we have chosen in Theorem 3.6 will now allow us tocompute the spectrum of the Laplacian on k -forms over the flat manifold. We will firstshow that this embedding implies that the essential spectrum contains a connectedinterval. HE SPECTRUM OF THE LAPLACIAN ON FORMS OVER FLAT MANIFOLDS 13
Proposition 4.1.
Let M = R n / Γ be a flat noncompact Riemannian manifold. Thenfor any ≤ k ≤ n , σ ess ( k, M ) ⊃ σ ess ( k, R n / Γ ) . Proof.
Note that R n / Γ = Z × R s where Z = V / Γ and V is the Euclidean spaceof dimension n − s . Let λ ∈ σ ess ( k, R n / Γ ). Then by the proof of Lemma 2.1, theapproximate eigenforms can be chosen as ω ∧ ρ e i √ µ r ω where ω is the first eigenform corresponding to the smallest eigenvalue λ of theLaplacian on ℓ -forms for some ℓ on Z = V / Γ , r is the distance function to the originon R s , ω = dx ∧ · · · ∧ dx k − ℓ , and ρ is the standard cut-off function so that theapproximation eigenform is of compact support within the annulus of radii R − R + 1 for R ≫ µ is chosen so that λ = λ + µ . We choose
C > R as in the proof of Theorem 3.6. Then by a standard integralestimate ρ e i √ µ r ω ∧ ω becomes the approximate eigenform for both R n / Γ and M for any nonnegative real number µ . This completes the proof of the proposition. (cid:3) We will now prove that the two spectra in Proposition 4.1 are in fact equal. Toachieve this, it suffices to show the following result.
Proposition 4.2.
Let λ o ( k, X ) denote the bottom of the Rayleigh quotient for theHodge Laplacian on k -forms over a Riemannian manifold X .The following inequalities hold λ o ( k, M ) ≥ λ o ( k, R n / Γ ) ,λ ess o ( k, M ) ≥ λ ess o ( k, R n / Γ ) = λ o ( k, R n / Γ ) . A key component of the proof of this proposition is the fact that the group Γ is ofpolynomial growth.
Proof.
To simplify notation we denote λ o ( k, M ) = λ o . Then for any ε >
0, thereexists a k -form ω ∈ C ∞ o (Λ k ( M )) such that R M |∇ ω | R M | ω | ≤ λ o + ε since the manifold is flat and the Weitzenb¨ock tensor on k -forms vanishes.We assume without loss of generality that R M | ω | = 1 . We consider the covering µ : R n / Γ → R n / Γ = M. If F = supp ( ω ) is the compact support of ω , we let { F j } be the lift of F in R n / Γ . We fix a point y ∈ R n / Γ and denote B y ( R ) the ball of radius R at y in R n / Γ . Let ξ ( R ) = { F j (cid:12)(cid:12) F j ∩ B y ( R ) = ∅ } . Let D be the diameter of F j . Let ρ be a cut-off function such that ρ = 1 on B y ( R ) ρ = 0 outside B y ( R ) |∇ ρ | ≤ CR − R , where R > R are big numbers.We consider the pull-back of the form ω , η = µ ∗ ω , on R n / Γ . Then the form ρη iscompactly supported in on R n / Γ . For any ε > Z R n / Γ |∇ ( ρη ) | ≤ (1 + ε ) Z R n / Γ ρ |∇ η | + (cid:18) ε (cid:19) Z R n / Γ |∇ ρ | | η | . Since µ is a local isometry we estimate Z R n / Γ ρ |∇ η | ≤ ξ ( R ) Z M |∇ ω | ≤ ξ ( R )( λ o + ε )and Z R n / Γ |∇ ρ | | η | ≤ ξ ( R ) C ( R − R ) Z M | ω | = ξ ( R ) C ( R − R ) . Combining the above we get Z R n / Γ |∇ ( ϕη ) | ≤ (cid:18) (1 + ε )( λ o + ε ) + (cid:18) ε (cid:19) C ( R − R ) (cid:19) ξ ( R ) . On the other hand, Z R n / Γ | ρη | ≥ ξ ( R − D ) Z M | ω | = ξ ( R − D ) . HE SPECTRUM OF THE LAPLACIAN ON FORMS OVER FLAT MANIFOLDS 15
Therefore, R R n / Γ |∇ ( ρη ) | R R n / Γ | ρη | ≤ (cid:18) (1 + ε )( λ o + ε ) + (cid:18) ε (cid:19) C ( R − R ) (cid:19) ξ ( R ) ξ ( R − D ) . By the Bishop-Gromov volume comparison theorem, we have R R n / Γ |∇ ( ρη ) | R R n / Γ | ρη | ≤ (cid:18) (1 + ε )( λ o + ε ) + (cid:18) ε (cid:19) C ( R − R ) (cid:19) (cid:18) R R − D (cid:19) n Choosing R , R sufficiently large with R /R →
1, and ε = ( R − R ) − →
0, weobtain λ o ( k, ∆ , R n / Γ ) ≤ λ o + o (1) . The proposition is proved. A similar method works for the essential spectrum. (cid:3)
As we mentioned in the proof of Theorem 3.6, the quotient Z = V / Γ is compact.The main theorem, Theorem 1.1 follows by combining this fact with Propositions 4.1,4.2 and Lemma 2.1. In fact, the bottom of the essential spectrum is determined bythe eigenvalues of the compact space Z = V / Γ depending on the order of the forms.Below we give a more detailed description of the bottom of the spectrum which is aconsequence of Lemma 2.1. Theorem 4.3.
Let M = R n / Γ be a flat noncompact Riemannian manifold. Let Z bethe compact flat manifold Z = V / Γ of dimension n − s . Then σ ( k, R n / Γ) = σ ess ( k, R n / Γ) = σ ess ( k, Z n − s × R s ) = [ α ( Z, s, n, k ) , ∞ ) where α ( Z, s, n, k ) is as in Lemma 2.1. The flatness of the manifold now gives the following immediate corollary.
Corollary 4.4.
The same result is true for the covariant Laplacian.
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Department of Mathematics and Statistics, University of Cyprus, Nicosia, 1678,Cyprus
E-mail address , Nelia Charalambous: [email protected]
Department of Mathematics, University of California, Irvine, Irvine, CA 92697,USA
E-mail address , Zhiqin Lu:, Zhiqin Lu: