Spectrum of generalized Hodge-Laplace operators on flat tori and ...

SPECTRUM OF GENERALIZED HODGE-LAPLACE OPERATORS ON FLAT TORI AND ROUND SPHERES STINE FRANZISKA BEITZ

arXiv:1510.08118v1 [math.DG] 27 Oct 2015

Abstract. We consider generalized Hodge-Laplace operators αdδ + βδd for α, β > 0 on p-forms on compact Riemannian manifolds. In the case of flat tori and round spheres of different radii, we explicitly calculate the spectrum of these operators. Furthermore, we investigate under which circumstances they are isospectral.

1. Introduction Can one hear the shape of a drum? Can one draw conclusions on its shape just based on its sound? This question was raised by Mark Kac already in 1966 ([Kac66]). However, he was not able to answer it completely. Mathematically, in his paper, a clamped elastic membrane is modeled by a domain G in the plane. The resonance frequencies are just the eigenvalues of the Dirichlet problem of the Laplacian ∆0 on functions, i.e. those real numbers λ, for which there are functions f : G → R, f 6≡ 0 which vanish on the boundary of G and comply with the eigenvalue equation ∆0 f = λf . In spectral geometry, similar problems are investigated in a more general setting. Here one is interested in the relationship between the geometric structures of Riemannian manifolds and the spectra of elliptic differential operators. In particular, one wonders which information the spectrum of these operators provides about the geometry of the underlying manifolds. One of the first results of this type was discovered by Hermann Weyl 1911 ([Wey11]). He showed that the volume of a bounded domain in the Euclidian space is determined by the asymptotic behaviour of the eigenvalues of the Dirichlet problem of the Laplace-Beltrami operator. To reconstruct a manifold completely up to isometry, the knowledge of the eigenvalues is, however, not sufficient. The answer to the question "Can One Hear the Shape of a Drum?" therefore is "no". This was already recognized by John Milnor, who proved the existence of two non-isometric 16-dimensional tori with identical spectra of the Laplacians ([Mil64]). Later, Gordon, Webb and Wolpert constructed different domains in the plane for which the eigenvalues coincide ([GWW92]). M In this paper, we consider the family of differential operators Fαβ := αdδ + βδd for real numbers 2 p ∗ 2 α, β > 0 on the Hilbert space L (M, Λ T M), i.e. the L p-forms on a compact Riemannian manifold (M, g) of dimension n with the Sobolev space H 2 (M, Λp T ∗ M) as domain of definition. Here d denotes the exterior derivative on differential forms and δ the adjoint operator of d. For α = β = 1 this yields the well-known Hodge-Laplace operator. Our aim is to determine the M M spectrum Spec(Fαβ ) of the operators Fαβ explicitly for certain manifolds and to investigate under which circumstances these operators are isospectral, i.e. possess the same spectra. M The investigation of the spectrum of the operator Fαβ finds applications, for example, in elasticity theory ([Zei88, p. 212]). Here the operator appears when computing the oscillation frequencies of an elastic body (for bodies made of simple materials). It turns out that these are determined by the eigenvalues of the operator in linear approximation. M The operator Fαβ is elliptic and self-adjoint on its domain of definition. Together with the compactness of M it follows (see for example [Gil95, Lemma 1.6.3]) that its spectrum is discrete and

Key words and phrases. Spectrum, differential operator, Hodge-Laplacian, compact manifold, torus, sphere, isospectrality, isometry. 1

2

STINE FRANZISKA BEITZ

only consists of real eigenvalues of finite multiplicities. Furthermore, the associated eigenforms M are smooth. Hence, to determine the spectrum of the operator Fαβ it suffices to investigate the p algebraic eigenvalue problem, i.e. to find solutions λ ∈ R and ω ∈ Ω (M) of the eigenvalue equaM M tion Fαβ ω = λω. Moreover, we just need to understand Fαβ as an operator on the smooth p-forms p Ω (M). In this work, we will calculate the spectrum, that is to say, the eigenvalues with the associated M eigenspaces, of the operators Fαβ for two sample-manifolds M − the spheres Srn of different radii r > 0 and the flat tori Rn /Λ induced by lattices Λ ⊂ Rn (see theorems 3.7 and 4.12). We will thereby discover that the spectrum splits into eigenvalues of the operators αdδ and βδd, which depend linearly on α and β, respectively. Proposition. If n = 2p, the spectrum is symmetric in α and β. T It will turn out that the eigenvalues of Fαβ on flat tori T are just αλ and βλ where λ are the eigenvalues of the ordinary Laplacian ∆0 on smooth functions. With this knowledge statements T1 about the isospectrality of two operators Fαβ and FαT′2β ′ for α, α′, β, β ′ > 0 on flat tori T1 and T2 can be made. First we will fix the coefficients α and β and notice that obviously for two compact isometric M N Riemannian manifolds M and N the operators Fαβ and Fαβ have the same spectrum. Then the question arises whether the converse of this statement in the case of flat tori holds as well, i.e. T whether the spectrum of Fαβ already determines the flat torus T up to isometry. In dimension n = 1, that is to say for 1-dimensional spheres, one directly sees that the answer is "yes". Next we prove for two flat tori T1 and T2 : T2 T1 are isospectral if and only if the Laplacians ∆T0 1 and ∆T0 2 and Fαβ Theorem. The operators Fαβ have the same spectrum.

Thereby we also can answer the above question for higher dimensions since the answer in the case of the Laplacian on functions is already known: In dimension n = 2 it is "yes" as well. For dimension n = 3 the question still seems to be open, even in the case of ∆0 . For the higher dimensions, we have the following result: Proposition. From dimension n = 4 onward there are non-isometric flat tori the spectra of which with respect to Fαβ coincide. On the other hand, fixing a flat torus T of dimension n 6= 2p and allowing arbitrary parameters α, α′ , β, β ′ > 0, we find that T Theorem. Fαβ and FαT′ β ′ can only be isospectral in the trivial case that these operators are already the same.

Finally, we consider two flat tori, the lattices of which are related by stretching by a factor c > 0. We show: Rn /Λ

Corollary. Fαβ

Rn /cΛ

and Fα′ β ′

have the same spectrum if and only if (α′ , β ′ ) = (c2 α, c2 β).

In dimension n = 2p, similar statements hold which take into consideration that the spectra of the operators Fαβ are symmetric in α and β, as already mentioned above. Rn /Λ Rn /Λ′ The question of what can be said about the spectra of Fαβ and Fα′ β ′ for arbitrary lattices Λ and Λ′ remains open. For n = 2, it is tempting to conjecture that these coincide if and only if there exists a c > 0 and a Q ∈ O(n) such that Λ′ = cQΛ and {α′ , β ′ } = {c2 α, c2 β}. In contrast to flat tori where the eigenvalue 0 always has the multiplicity np , the eigenvalues on Sn

Sn

spheres are all positive. Here again, we compare the operators Fαβr and Fα′rβ′ ′ .

SPECTRUM OF GENERALIZED HODGE-LAPLACE OPERATORS ON TORI AND SPHERES

3

Proposition. If these operators have the same spectra, the radii r and r ′ are equal if and only if α = α′ and β = β ′ (for n = 2p, up to exchange of the roles of α and β). For c > 0 such that r ′ = cr we can show: Proposition. The isospectrality of the two above mentioned operators is equivalent to (α′, β ′ ) = (c2 α, c2β) (for n = 2p, again up to exchange of the roles of α and β). Acknowledgements. I would like to express my special thanks to Prof. Dr. Christian Bär for proposing the topic of this paper, for lots of stimulating conversations and many useful hints and references. It is my pleasure to thank Dr. Andreas Hermann for his advice and many valuable suggestions. Moreover, I am grateful to Matthias Ludewig for his support and for numerous helpful discussions. 2. Preliminaries Let M be an n-dimensional Riemannian manifold, α, β > 0 and 0 ≤ p ≤ n.

Throughout the paper, we write Ωp (M) := Γ(M, Λp T ∗ M) for the space of all smooth differential forms of degree p on M and, for its complexification, we write Ωp (M, C) := Ωp (M) ⊗ C. We denote by ΩpL2 (M) := L2 (M, Λp T ∗ M) the completion of the smooth p-forms with compact support Ωpc (M) with respect to the L2 -scalar product (·, ·). Furthermore, we write ΩpL2 (M, C) := ΩpL2 (M) ⊗ C for its complexification. Now we introduce the central object of this paper. Definition 2.1. We define the operator M Fαβ,p := αdδ + βδd

on the space Ωp (M). Here, d is the exterior derivative on differential forms and δ the formal adjoint of d, i.e. for all ω ∈ Ωpc (M) and η ∈ Ωp+1 c (M) we have (dω, η) = (ω, δη). We simply write M Fαβ if it is clear from the context on which forms the operator is considered. M The Hodge-Laplace operator on Ωp (M) we denote by ∆M p := F11,p = dδ + δd. Remark 2.2. M i) The principal symbol of Fαβ is −β|ξ|2id − (α − β)ξ ∧ (ξ ♯ y ·),

where ξ ∈ T ∗ M. Here ξ ♯ yω := ω(ξ ♯ , ·, ..., ·) for ω ∈ Ωp (M) and ♯ denotes the musical isomorphism which assigns to each ξ ∈ T ∗ M the uniquely determined vector ξ ∈ T M such that g(ξ ♯ , ·) = ξ. For each ξ 6= 0, the principal symbol is invertible with inverse map 1 α−β − id + ξ ∧ (ξ ♯ y ·). 2 4 β|ξ| αβ|ξ| M This shows that Fαβ is elliptic. M ii) Fαβ is formally self-adjoint. M iii) Any two eigenforms to different eigenvalues of Fαβ are orthogonal with respect to the 2 L -scalar product.

Remark 2.3. The exterior differential d and the co-derivative δ can be written in terms of the connection on forms in the following way: Let {e1 , ..., en } be a local orthonormal basis of T M and {e1 , ..., en } the associated dual basis of T ∗ M. Then n n X X i d= e ∧ ∇ei and δ = − ei y∇ei . i=1

i=1

4


M In order to understand the spectrum of Fαβ (in the case that M is compact) as a set of eigenvalues with associated multiplicities, we introduce the concept of a weighted set.

Definition 2.4. i) A weighted set is a function W : C → N0 . ii) If W has a countable support supp(W ) := {λ ∈ C | W (λ) 6= 0} = {λi | i ∈ N}, we write W := {(λ1 , W (λ1)), (λ2 , W (λ2 )), ...}

and respectively

W := { λ1 , ..., λ1 , λ2 , ..., λ2 , ...}W . | {z } | {z } W (λ1 )-times W (λ2 )-times

iii) Let W and W ′ be weighted sets. Then their weighted union W ⋒ W ′ is defined as: (W ⋒ W ′ )(λ) := W (λ) + W ′ (λ) for all λ ∈ C. iv) In addition, we introduce the following notation for m, m′ ∈ N0 : W

m

′

⋒m W ′ := W ... ⋒ W} ⋒ |W ′ ⋒ {z ... ⋒ W}′ . | ⋒ {z m -times

m′ -times

For m = 1 or m′ = 1 we usually omit the index. v) The difference of W and W ′ is the weighted set W \ W ′ , which is defined by (W \ W ′ )(λ) := max{W (λ) − W ′ (λ), 0}

for all λ ∈ C. vi) The minimum of a weighted set W with support supp(W ) ⊆ R is given by min(W ) := min supp(W ) . vii) For r ∈ R∗ let rW (λ) := W ( λr ) for all λ ∈ C. Unlike in ordinary sets, in weighted sets elements can appear several times. Definition 2.5. Let M be compact. i) We call M M Eig(Fαβ,p , λ) := {ω ∈ Ωp (M) | Fαβ,p ω = λω}

M the eigenspace of Fαβ,p to the eigenvalue λ. M ii) The spectrum of Fαβ,p is the weighted set for λ ∈ C defined by M M Spec(Fαβ,p )(λ) := dim Eig(Fαβ,p , λ) .

Remark 2.6. The measure

X

M Spec(Fαβ,p )(λ)δλ ,

λ∈C

M where δλ is the Dirac measure at λ ∈ C, is exactly the spectral measure of Fαβ,p .

Proposition 2.7. Let M be an n-dimensional orientable Riemannian manifold, α, β > 0 and M M 0 ≤ p ≤ n. Then the spectra of Fαβ,p und Fβα,n−p coincide. Proof. Let ∗p : Ωp (M) → Ωn−p (M) be the Hodge-Star operator on p-forms. Using the properties one can show that

∗n−p ∗p ω = (−1)p(n−p) ω

and δ = (−1)p+1 ∗p+1 d∗−1 p ,

M M ∗p Fαβ,p ∗−1 p = Fβα,n−p .


5

M Let now ω ∈ Eig(Fαβ,p , λ). Then

M M Fβα,n−p ∗p ω = ∗p Fαβ,p ω = ∗p λω = λ ∗p ω,

M M i.e. ∗p ω ∈ Eig(Fβα,n−p , λ). Conversely, if η ∈ Eig(Fβα,n−p , λ) ⊆ Ωn−p (M), because of the bijectivity of ∗p , there is an ω ∈ Ωp (M) with η = ∗p ω and we have M M M ∗p Fαβ,p ω = ∗p Fαβ,p ∗−1 p η = Fβα,n−p η = λη = λ ∗p ω = ∗p λω.

M M Hence Fαβ,p ω = λω and therefore ω ∈ Eig(Fαβ,p , λ). Thus, in total, we have shown that the map : Eig(F M , λ) → Eig(F M ∗p M Eig(Fαβ,p ,λ)

αβ,p

βα,n−p , λ)

is bijective and, as a result,

M M Spec(Fαβ,p ) = Spec(Fβα,n−p ).

3. Spectrum on flat tori In this section, let α, β > 0 and 1 ≤ p ≤ n. We will investigate and determine explicitely the Tn spectrum of Fαβ,p on n-dimensional flat tori T n . Definition 3.1. Let B := {b1 , ..., bn } be a basis of Rn and {b1 , ..., bn } the associated dual basis of (Rn )∗ , i.e. bi (bj ) = δji for all i, j ∈ {1, ..., n}. Then ΛB := Zb1 + ... + Zbn

is the lattice induced by B and Λ∗B := Zb1 + ... + Zbn = {l ∈ (Rn )∗ | ∀λ ∈ Λ : l(λ) ∈ Z}

the dual lattice of ΛB .

Let Λ be a lattice in Rn and π : Rn → Rn /Λ : x 7→ [x] the canonical projection on Rn /Λ. Then (Rn /Λ, g) is an n-dimensional flat torus associated to the lattice Λ, which we often denote by T n , where Rn /Λ is endowed with the Riemannian metric g induced by the standard metric gstd on Rn via gstd = π ∗ g. We remark that flat tori are Lie groups with respect to the usual addition. We write {∂1 , ..., ∂n } and {dx1 , ..., dxn } for the global bases of T T n und T ∗ T n induced by the standard basis of Rn . Due to the existence of a global basis of T ∗ T n , for flat tori T n we have the following isomorphism: (1)

L2 (T n , C) ⊗ Λp (Rn )∗ −→ ΩpL2 (T n , C) n n X X i1 ip fi1 ...ip e ∧ ... ∧ e 7−→ fi1 ...ip dxi1 ∧ ... ∧ dxip ,

i1 ,...,ip =1

i1 ,...,ip =1

where {e , ..., e } is the dual basis of the standard basis in Rn . 1

n

Definition 3.2. Let {e1 , ..., en } be the dual basis of the global standard basis of T Rn and T n a flat torus. Then Ωppar (T n ) := {ω ∈ Ωp (T n ) | ∇ω = 0}  n  X = ai1 ...ip dxi1 ∧ ... ∧ dxip  i1 ,...,ip =1

and

  ai1 ...ip ∈ R 

  ip i1 n p Ωpar (R ) := ai1 ...ip e ∧ ... ∧ e ai1 ...ip ∈ R   i1 ,...,ip =1  n  X

6


are the spaces of all parallel p-forms on T n and Rn respectively. Furthermore let Ωppar (T n , C) := Ωppar (T n ) ⊗ C and Ωppar (Rn , C) := Ωppar (Rn ) ⊗ C. Remark 3.3. For flat tori T n , we have

by virtue of

Ωppar (T n ) ∼ = Λp (Rn )∗ n n X X i1 ip ai1 ...ip dx ∧ ... ∧ dx 7−→ ai1 ,...,ip ei1 ∧ ... ∧ eip ,

i1 ,...ip =1

i1 ,...,ip =1

where {e1 , ..., en } is the dual basis of the standard basis of Rn . We will identify both vector spaces without always stating it explicitly. We need to know how δ acts on Ωp (T n , C)P for flat tori T n with respect to the bases induced by the dxi , i ∈ {1, ..., n}. To this end, let ω = ni1 ,...,ip=1 ωi1 ...ip dxi1 ∧ ... ∧ dxip ∈ Ωp (T n , C). Then one can check that (2)

p n X X d ik ∧ ... ∧ dxip . (−1)k−1 ∂ik ωi1 ...ip dxi1 ∧ ... ∧ dx δω = − i1 ,...,ip =1 k=1

This is also true when T n and {dx1 , ..., dxn } are replaced by Rn and the dual basis {e1 , ..., en } of the global standard basis of T Rn , respectively. Remark 3.4. Let T n := Rn /Λ be the flat torus to the lattice Λ. i) For l ∈ Λ∗ the funcions χl : T n → C : [x] 7→ e2πil(x) are exactly the characters of the Lie group T n and form a basis of L2 (T n , C), according to the Peter-Weyl theorem ([Wer04, p. 250]). The χl are well defined, since for all l ∈ Λ∗ the map Rn → C : x 7→ e2πil(x) is Λ-periodic, more precisely: For λ ∈ Λ, we have χl ([x + λ]) = e2πil(x+λ) = e2πil(x) e2πil(λ) = e2πil(x) = χl ([x]) since l(λ) ∈ Z. ii) For l ∈ Λ∗ the χl (and multiples of it) are exactly the eigenfunctions of ∆0 on C ∞ (T n , C) to the eigenvalues λl1 := 4π 2 |l|2 and we have n

Spec(∆T0 ) = ⋒ ∗ {λl1 }1 .

(3)

l∈Λ

3.1. Eigendecomposition. In this section, let Λ be a lattice in Rn , n > 0 and T n := Rn /Λ the Tn together associated flat torus. For α, β > 0 and 1 ≤ p ≤ n we calculate the eigenvalues of Fαβ,p with the associated eigenspaces. P Let ω = ni1 ,...,ip=1 ωi1 ,...,ip dxi1 ∧...∧dxip ∈ Ωp (T n , C). Then for all i1 , ..., ip ∈ {1, ..., n} the ωi1 ...ip ∈ C ∞ (T n , C) ⊂ L2 (T n , C). Thus, because of Remark 3.4, for each l ∈ Λ∗ there are ωil1 ...ip ∈ C, such that the series X ωil1 ...ip χl ωi1 ...ip = l∈Λ∗

converges in L2 (T n , C). We set l

ω :=

n X

i1 ,...,ip =1

ωil1 ...ip dxi1 ∧ ... ∧ dxip ∈ Ωppar (T n , C) ∼ = Λp (Cn )∗ .


Hence ω =

P

l∈Λ∗

χl ω l . We have

dδω = − =−

n X

p X

i1 ,...,ip ,k=1 j=1 n X

i1 ,...,ip

= 4π 2

d ij ∧ ... ∧ dxip (−1)j−1∂k ∂ij ωi1 ...ip dxk ∧ dxi1 ∧ ... ∧ dx

p X X

dij ∧ ... ∧ dxip (−1)j−1 ωil1 ...ip ∂k ∂ij χl dxk ∧ dxi1 ∧ ... ∧ dx | {z } ,k=1 j=1 l∈Λ∗

p n X X X

i1 ,...,ip =1 j=1 l∈Λ∗

= 4π 2

X

=(2πi)2 lk lij χl

d ij ∧ ... ∧ dxip (−1)j−1ωil1 ...ip lij χl l ∧ dxi1 ∧ ... ∧ dx

χl l ∧ (l♯ yω l )

l∈Λ∗

and analogously

7

δdω = 4π 2

X

l∈Λ∗

Put together, we obtain n

T Fαβ ω = 4π 2

X

l∈Λ∗

χl − l ∧ (l♯ yω l ) + |l|2 ω l .

χl (α − β)l ∧ (l♯ yω l ) + β|l|2ω l .

Due to Remark 3.4, comparison of coefficients gives that the eigenvalue equation X Tn Fαβ ω = λω = λ χl ω l l∈Λ∗

is satisfied for a λ ∈ R if and only if for all l ∈ Λ∗ , we have (EGλ,l ) 4π 2 (α − β)l ∧ (l♯ yω l ) + β|l|2 ω l = λω l . Definition 3.5. For l ∈ Λ∗ and γ ∈ R, we set

λlγ := 4π 2 γ|l|2

and n Vlp := χl {l ∧ η | η ∈ Ωp−1 par (T )},

Wlp := χl {ω ∈ Ωppar (T n ) | l♯ yω = 0}.

Remark 3.6. It is obvious that for all k, l ∈ Λ∗ with k 6= l the spaces Vkp , Vlp , Wkp and Wlp are pairwise orthogonal. Therefore Vkp ⊕ Vlp , Wkp ⊕ Wlp , Vkp ⊕ Wlp und Vkp ⊕ Wkp are direct sums. n ♯ Now let l ∈ Λ∗ . We notice that l ∧ η, with η ∈ Ωp−1 par (T ) and l yη = 0, satisfies the equation (EGλlα ,l ) and ω l ∈ Ωppar (T n ), with l♯ yω l = 0, the equation (EGλlβ ,l ). Therefore, we have n

T Fαβ χl l ∧ η = λlα χl l ∧ η

n

T and Fαβ χl ω l = λlβ χl ω l . n

T Hence, if η, ω l 6= 0, then χl l ∧ η and χl ω l are eigenforms of Fαβ to the eigenvalues λlα and λlβ , respectively. Because

n−1 dim {l ∧ η | η ∈ = p−1 n−1 p n ♯ , dim {ω ∈ Ωpar (T , C) | l yω = 0} = p n Ωp−1 par (T , C)}

8


n−1

where we use the convention n := 0, we already know that Tn l l ) ⋒ ∗ {λα }(n−1) ⋒ ⋒ ∗ {λβ }(n−1) ⊆ Spec(Fαβ,p l∈Λ

and that, for all l ∈ Λ∗ , (4)

l∈Λ

p−1

p

n

n

T T Vlp ⊆ Eig(Fαβ,p , λlα ) and Wlp ⊆ Eig(Fαβ,p , λlβ ).

The following theorem shows that with λlα and λlβ for l ∈ Λ∗ , we already found the whole spectrum Tn . of Fαβ n

T is given by Theorem 3.7. Let α, β > 0 and 1 ≤ p ≤ n. The spectrum of the operator Fαβ,p l Tn l Spec(Fαβ,p ) = ⋒ ∗ {λα }(n−1) ⋒ ⋒ ∗ {λβ }(n−1) p p−1 l∈Λ l∈Λ

and the associated eigenspaces are for k ∈ Λ∗ n

T Eig(Fαβ,p , λkα ) =

(5)

M

Vlp ⊕

M

Wlp

M

Vl p ⊕

M

Wlp .

l∈Λ∗ : |l|=|k|

and n

T Eig(Fαβ,p , λkβ ) =

(6)

l∈Λ √∗ : |l|= α |k| β

l∈Λ √∗β: |k| |l|= α

l∈Λ∗ : |l|=|k|

Proof. Let {ei }i∈{1,...,n} ⊆ (Rn )∗ be the dual basis of the standard basis of Rn . Because {χl }l∈Λ∗ is a basis of L2 (T n , C), the vectors {χl ei1 ∧...∧eip }l∈Λ∗, 1≤i1 0 and k ∈ Λ∗ as above. We set 2p−1 ( p ) e ′ f′ := M ′ \ {(0, 2p−1 )}. We have m f′ ) > 0 γ := λm γA and M e ′ := min(M k . Now let M := M \ ⋒i=1 p 1 2p−1 2p−1 e′ and we define δ := m . Then M = γA ( p ) ⋒ ( p ) δA, where {γ, δ} is unique by construction. λk1


13

2p T 2p ) = Spec(FαT′ β ′ ,p ) = feA ({α′ , β ′ }) we obtain that Thus feA is injective. With feA ({α, β}) = Spec(Fαβ,p {α, β} = {α′ , β ′ }.

"⇐": The converse implications are trivial.

The next corollary shows that there are α, α′, β, β ′ > 0 and lattices Λ, Λ′ in Rn such that Rn /Λ′

Rn /Λ Fαβ,p

and Fα′ β ′ ,p are isospectral. Corollary 3.21. Let Λ be a lattice in Rn , α, α′ , β, β ′ > 0 and c ∈ R \ {0}. Then for n 6= 2p: Rn /Λ Rn /cΛ Fαβ,p and Fα′ β ′ ,p are isospectral if and only if (α′, β ′ ) = (c2 α, c2β). For n = 2p the statement holds with {α′ , β ′ } = {c2 α, c2 β} instead of (α′ , β ′) = (c2 α, c2 β). Rn /cΛ

Proof. For the spectrum of Fα′ β ′ ,p , we have n R /cΛ Spec Fα′ β ′ ,p = α′

α′ = 2 c

2

2

⋒ {4π |l| }1

l∈ 1c Λ∗

2

2

⋒ {4π |l| }1 Rn /Λ = Spec F α′ β ′ . l∈Λ∗

c2 c2

!

(

n−1 p−1

) ⋒(

)β

n−1 p

′

n−1 β (n−1 p−1 ) ⋒ ( p ) c2

′

2

2

⋒ {4π |l| }1

l∈ 1c Λ∗

2

2

⋒ {4π |l| }1

l∈Λ∗

!

,p

Rn /Λ

Rn /cΛ

Theorem 3.20 therefore implies that Fαβ,p and Fα′ β ′ ,p are isospectral in dimension n 6= 2p if and only if (α′ , β ′) = (c2 α, c2 β) and in dimension n = 2p if and only if {α′ , β ′ } = {c2 α, c2 β}. 4. Spectrum on round spheres Now we consider for n ∈ N the n-dimensional unit sphere (S n , g) ⊂ (Rn+1 , gstd ), which is embedded in Rn+1 via the canonical inclusion ι : S n → Rn+1 and endowed with the metric g := ι∗ gstd induced by the standard metric gstd (·, ·) := h·, ·i on Rn+1 . In the following, let α, β > 0 and 1 ≤ p ≤ n. e := X n the restriction of X to For vector fields X ∈ Γ(Rn+1 , T Rn+1 ) on Rn+1 we denote by X S S n . Note that this is a vector field on S n if and only if X is tangent to S n . From now on, let ∇ e the one on S n . The outward directed normal vector be the Levi-Civita connection on Rn+1 and ∇ field we call x ν : Rn+1 \ {0} → T Rn+1 : x 7→ . kxk Its covariant derivative with respect to X ∈ Γ(Rn+1 , T Rn+1) is 1 (8) ∇X ν = X − hX, νiν . r n+1 Here, r : R → R≥0 is defined by x 7→ r(x) := kxk. e are related as follows: For X, Y ∈ Γ(Rn+1 , T Rn+1 ) tangent to S n , we The connections ∇ and ∇ have e e ν. e e ] (9) ∇ X Y = ∇ e Y − hX, Y ie X

On can easily convince oneself of the fact that (10)

f1, ..., X fp ) ι∗ (ω(X1, ..., Xp )) = (ι∗ ω)(X

fi ) = X fi for for all ω ∈ Ωp (Rn+1 ) and all X1 , ..., Xp ∈ Γ(Rn+1 , T Rn+1) tangent to S n , since dι(X i ∈ {1, ..., p}.

14


The exterior derivative d is natural, i.e. commutes with the pullback f ∗ along differentiable maps f . In general, this is not true for the co-differential δ instead of d. In the following lemma, we investigate in which way δ commutes with the pullback ι∗ : Ω∗ (Rn+1 ) → Ω∗ (S n ). Lemma 4.1. Let ω ∈ Ωp (Rn+1 ). Then ι∗ δ R

n+1

n ω = δ S ι∗ ω − ι∗ (n − p + 1) · (νyω) + νy∇ν ω .

Proof. Let X, Y1 , ..., Yp be vector fields on Rn+1 tangent to S n . Due to the naturality of d and the formulas (9) and (10), we have (ι∗ ∇X ω)(Ye1, ..., Yep ) = ι∗ (∇X ω)(Y1, ..., Yp ) ! p X ω(Y1, ..., Yi−1 , ∇X Yi , Yi+1, ..., Yp ) = ι∗ ∂X ω(Y1, ..., Yp ) − i=1 p

X ∗ e ^ g = ∂Xe (ι∗ ω)(Ye1 , ..., Yep )) − (ι ω)(Ye1, ..., Yg i−1 , ∇X Yi , Yi+1 , ..., Yp ) i=1

= ∂Xe (ι∗ ω)(Ye1 , ..., Yep )) − +

p X i=1

p X i=1

e e Yei , Yg e (ι∗ ω)(Ye1, ..., Yg i−1 , ∇X i+1 , ..., Yp )

e Yei i(ι∗ ω)(Ye1, ..., Yg e hX, e, Yg i−1 , ν i+1 , ..., Yp ) p X

e e ι∗ ω)(Ye1, ..., Yep ) + ι∗ = (∇ X

(−1)i−1 X ♭ (Yi )ω(ν, Y1, ..., Yi−1 , Yi+1 , ..., Yp )

i=1

e e ι∗ ω)(Ye1, ..., Yep ) + ι∗ (X ♭ ∧ (νyω)) (Ye1 , ..., Yep ), = (∇ X

i.e.

!

e e ι∗ ω + ι∗ X ♭ ∧ (νyω) . ι∗ ∇X ω = ∇ X

Here ♭ denotes the inverse of the musical isomorphism ♯. Let U ⊆ S n be open and {e˜1 , ..., e˜n } ⊂ Γ(U, T S n ) a local orthonormal basis of T S n . We set y V := {y ∈ Rn+1 | kyk ∈ U}. For i ∈ {1, ..., n} let ei ∈ Γ(V, T Rn+1) be the radial constant y ) for y ∈ V . Then {e1 , ..., en , ν} is a extension of e˜i on V ⊂ Rn+1 , i.e. defined via ei (y) := e˜i ( kyk n+1 local orthonormal basis of T R . With the above calculation it follows that ι∗ δ

Rn+1

ω = −ι∗ =− =−

n X

i=1 n X i=1

Sn ∗

n X

ei y∇ei ω

i=1

!

− ι∗ νy∇ν ω

eei yι∗ ∇ei ω − ι∗ νy∇ν ω e ee ι∗ ω − eei y∇ i ∗

=δ ι ω−ι

n X i=1

n X i=1

eei yι∗ ei ∧ (νyω) − ι∗ (νy∇ν ω)

i

ei y e ∧ (νyω) + νy∇ν ω

!

n = δ S ι∗ ω − ι∗ (n − p + 1) · (νyω) + νy∇ν ω .


15

n

S 4.1. Eigendecomposition. To determine the spectrum Spec(Fαβ ), we first investigate the relan+1 n R S tion between the operators Fαβ and Fαβ .

Proposition 4.2. For ω ∈ Ωp (Rn+1 ), we have ∗ ∗ Rn+1 Sn ∗ ι Fαβ ω = Fαβ ι ω + αι (p − n − 1)d νyω − d νy∇ν ω + βι∗ (n − p − 1)d νyω + d νy∇ν ω + p(p − n + 1)ω − n∇ν ω − ∇ν ∇ν ω . Proof. Due to the naturality of d and Lemma 4.1, we see that (11)

ι∗ dδ R

n+1

ω = dι∗ δ R

and ι∗ δ R

(12)

n+1

n+1

n

ω = dδ S ι∗ ω − ι∗ (n − p + 1)d(νyω) + d(νy∇ν ω)

n dω = δ S dι∗ ω − ι∗ (n − p) · (νydω) + νy∇ν dω .

Now we calculate the last two terms of (12). To this end, let {e1 , ..., en , en+1 := ν} be a local orthonormal basis of T Rn+1 as in the proof of Lemma 4.1. Then νydω =

n X i=1

=

νy(ei ∧ ∇ei ω) + νy(ν ♭ ∧ ∇ν ω)

n X i=1

ei (ν) ∇ei ω − ei ∧ (νy∇ei ω) + ν ♭ (ν) ∇ν ω − ν ♭ ∧ (νy∇ν ω). | {z } | {z } =0

=1

Due to ι∗ (ν ♭ ) = 0, ι∗ (r) = 1 and

1 νy∇ei ω = ∇ei (νyω) − (∇ei ν)yω = ∇ei (νyω) − ei yω r for i ∈ {1, ..., n}, this yields (14) ι∗ (νydω) = ι∗ − d(νyω) + pω + ∇ν ω . (13)

We notice that ∇ν ei = 0 for i ∈ {1, ..., n + 1}, which is why with (8) it follows that [ei , ν] = 1r ei . Therefore we see that for i ∈ {1, ..., n} (15)

1 1 (13) νy∇ν ∇ei ω = ∇ν (νy∇ei ω) = ∇ν ∇ei (νyω) + 2 ei yω − ∇ν (ei yω) | {z } r r =∇ei ∇ν −∇[ei ,ν]

1 1 1 = ∇ei ∇ν (νyω) − ∇ei (νyω) + 2 ei yω − ei y∇ν ω. r r r

Since ∇ν ei = 0, we have νy∇ν dω =

n+1 X i=1

=

n X i=1

=

n X i=1

=−

i

νy∇ν (e ∧ ∇ei ω) =

n+1 X i=1

νy(∇ν ei ∧∇ei ω + ei ∧ ∇ν ∇ei ω) | {z } =0

νy(ei ∧ ∇ν ∇ei ω) + νy(ν ♭ ∧ ∇ν ∇ν ω)

ei (ν)∇ν ∇ei ω − ei ∧ (νy∇ν ∇ei ω) + ν ♭ (ν) ∇ν ∇ν ω − ν ♭ ∧ (νy∇ν ∇ν ω) | {z } | {z }

n X i=1

=0

=1

ei ∧ (νy∇ν ∇ei ω) + ∇ν ∇ν ω − ν ♭ ∧ (νy∇ν ∇ν ω).

16


If we apply the pullback to it, using (15), ι∗ (ν ♭ ) = 0 and ι∗ (r) = 1 gives ι∗ (νy∇ν dω) = ι∗ (−d(∇ν (νyω)) + d(νyω) − pω + p∇ν ω + ∇ν ∇ν ω).

(16)

Inserting (14) and (16) into (12), we obtain n+1 n ι∗ δ R dω = δ S dι∗ ω + ι∗ (n − p − 1)d(νyω) + p(p − n + 1)ω − n∇ν ω + d(νy∇ν ω) − ∇ν ∇ν ω . Together with (11), the proposition follows.

Corollary 4.3. For ω ∈ Ωp (Rn+1 ), we have n+1 n ι∗ (∆R ω) = ∆S (ι∗ ω) + ι∗ p(p − n + 1)ω − 2d(νyω) − n∇ν ω − ∇ν ∇ν ω .

Proof. This is Proposition 4.2 for α = β = 1.

Definition 4.4. Let {e1 , ..., en+1 } be the global standard basis of T Rn+1 and {e1 , ..., en+1 } the associated dual basis. For each k ∈ N0 we define Hk0 := {P ∈ C ∞ (Rn+1 ) | P is a homogeneous polynomial of degree k with ∆R0 and the space   p Hk := ω = 

n+1

P = 0}

0 Rn+1 i1 ip p n+1 ω=0 ωi1 ...ip e ∧ ... ∧ e ∈ Ω (R ) ωi1 ...ip ∈ Hk and δ 1≤i1 0 and 1 ≤ p ≤ n. Then Fαβ,p and Fαβ,p ′ only if r = r .

Sn

Sn

r r′ Proof. "⇒": Since Fαβ,p and Fαβ,p have the same spectrum, in particular their smallest eigenvalues (p+1)(n−p) α coincide. In the case that β ≥ p(n−p+1) , we have that λ1β,p,r ≤ µ0α,p,r and λ1β,p,r′ ≤ µ0α,p,r′ . It follows

Sn

Sn

r′ r )) if and only if β(p+1)(n−p) = λ1β,p,r = λ1β,p,r′ = β(p+1)(n−p) , )) = min(Spec(Fαβ,p that min(Spec(Fαβ,p r2 r ′2 (p+1)(n−p) α ′ 1 0 1 0 i.e. if and only if r = r . If β < p(n−p+1) , we have λβ,p,r > µα,p,r and λβ,p,r′ > µα,p,r′ . Hence

Sn

Sn

r′ r )) if and only if )) = min(Spec(Fαβ,p min(Spec(Fαβ,p ′ and only if r = r . "⇐": This direction is trivial.

αp(n−p+1) r2

= µ0α,p,r = µ0α,p,r′ =

αp(n−p+1) , r ′2

i.e. if

22


4.3.2. Variation of parameters. We now discuss the question how the spectra of two operators Srn′ Srn Fαβ,p und Fα′ β ′ ,p′ are related for different parameters α, α′ , β, β ′, r, r ′ > 0 und 1 ≤ p, p′ ≤ n. As a first step, we will fix the radii of the spheres. Remark 4.19. Proposition 2.7 implies that for all α, β, r > 0 and n = 2p Sn

Sn

r r ). ) = Spec(Fβα,p Spec(Fαβ,p

Sn

Sn

r and Fα′rβ ′ ,p are Theorem 4.20. Let α, α′, β, β ′, r > 0 and 1 ≤ p ≤ n with n 6= 2p. Then Fαβ,p isospectral if and only if (α, β) = (α′, β ′ ). For n = 2p the statement holds with {α, β} = {α′ , β ′ } instead of (α, β) = (α′ , β ′ ).

Proof. "⇒": Let at first n 6= 2p. We show that the map

Sn

r ) f : (0, ∞) × (0, ∞) → M : (γ, δ) 7→ Spec(Fγδ,p

is injective. To this end, let M be in the image of f . Due to Remark 4.13, we have m := min(M) > 0. Furthermore, due to Theorem 4.12 and Proposition 4.16, the multiplicity mult(m) of m satisfies n+2 n+1 n+1 4.14 p p p p . , , mult(m) ∈ {dim(V1 ), dim(W0 ), dim(V1 ) + dim(W0 )} = p+1 p p+1 Since 1 ≤ p ≤ n and n 6= 2p, the set in fact consists of three different elements. • In the case that mult(m) = dim(V1p ), we set δ :=

r2 m . (p+1)(n−p)

r 2 m′ . p(n−p+1) 2 r m and p(n−p+1)

and m′ := min(M ′ ). Then we define γ :=

Let M ′ := M\⋒k∈N {λkδ,p,r }dim(Vkp ) 2

r m δ := (p+1)(n−p) . Here M ′ := M \ • If mult(m) = dim(W0p ), we put γ := ⋒k∈N0 {µkγ,p,r }dim(Wkp ) and m′ := min(M ′ ). r2 m r2 m • In the last case that mult(m) = dim(V1p ) + dim(W0p ) let γ := p(n−p+1) and δ := (p+1)(n−p) . ′

Sn

r ). Here γ and δ are unique by construction. Thus f is injective. In each case, M = Spec(Fγδ,p Since by assumption f (α, β) = f (α′ , β ′), it follows that (α, β) = (α′ , β ′ ).

Now let n = 2p. We consider the map Sr2p fe : C ⊂ (0, ∞) | #C ∈ {1, 2} → M : {γ, δ} 7→ Spec(Fγδ,p ). Let M be in its image. Then m := min(M) > 0. We set γ := 2

′

r2 m . p(p+1)

Let M ′ := M \ S2

r m r ⋒k∈N0 {µkγ,p,r }dim(Wkp ) and m′ := min M ′ . We define δ := p(p+1) ), whereat . Then M = Spec(Fγδ,p {γ, δ} is unique by construction. Consequently, fe is injective. Hence, fe({α, β}) = fe({α′ , β ′ }) implies that {α, β} = {α′, β ′ }. "⇐": The opposite directions are trivial.

For different radii, the spectra are related as follows: Sn

Sn

r and Fα′rβ′ ′ ,p are isospectral Corollary 4.21. Let α, α′ , β, β ′, r, r ′ > 0 and 1 ≤ p ≤ n such that Fαβ,p and n 6= 2p. Then r = r ′ if and only if (α, β) = (α′ , β ′ ). For n = 2p the statement holds with {α, β} = {α′ , β ′ } instead of (α, β) = (α′ , β ′ ).

Proof. The first direction is just Theorem 4.20 and the opposite direction Proposition 4.18. Sn

Sn

r Proposition 4.22. Let α, α′, β, β ′, r, c > 0, 1 ≤ p ≤ n and n 6= 2p. Then Fαβ,p and Fα′crβ ′ ,p are isospectral if and only if (α′ , β ′ ) = (c2 α, c2 β). For n = 2p the statement holds with {α′, β ′ } = {c2 α, c2 β} instead of (α′ , β ′ ) = (c2 α, c2 β).

SPECTRUM OF GENERALIZED HODGE-LAPLACE OPERATORS ON TORI AND SPHERES Sn

23

Sn

r and Fα′crβ ′ ,p are isospectral if and only if Proof. Proposition 4.16 tells us that Fαβ,p n 1 Scr Srn Sn 2 2 Sn Sn Spec(Fαβ,p ) = r Spec(Fαβ,p ) = r Spec(Fα′ β ′ ,p ) = 2 Spec(Fα′ β ′ ,p ) = Spec F α′ β ′ ,p . c c2 c2 ′ ′ Due to Theorem 4.20, for n 6= 2p this is equivalent to (α, β) = αc2 , cβ2 , and for n = 2p to n ′ ′o {α, β} = αc2 , βc2 .

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