Table Of ContentPeriodic Manifolds, Spectral Gaps, and
Eigenvalues in Gaps
@
VomFachbereich fu¨rMathematikundInformatik
derTechnischenUniversita¨tBraunschweig
genehmigte
Dissertation
zur ErlangungdesGrades eines
DoktorsderNaturwissenschaften
(Dr.rer. nat.)
von
Olaf Post
Braunschweig, Juli2000
Eingereicht am 30. Mai 2000
Datum der mu¨ndlichen Pru¨fung: 13. Juli 2000
1. Referent: Prof. Dr. Rainer Hempel
2. Referent: Priv.-Doz. Dr. Norbert Knarr
Fu¨r Claudia
Contents
Introduction 3
1 Preliminaries 11
1.1 Hilbert spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.2 Operators and quadraticforms . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3 Spectrum and Min-maxPrinciple . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.4 Parameter-dependent Hilbert spaces . . . . . . . . . . . . . . . . . . . . . . . 15
1.5 Interchange ofnorm andquadraticform . . . . . . . . . . . . . . . . . . . . . 20
2 Analysisonmanifolds 23
2.1 Spaces ofsquareintegrablefunctions . . . . . . . . . . . . . . . . . . . . . . . 23
2.2 Sobolevspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.3 TheLaplacian on amanifold . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.4 Metricperturbations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.5 TheDecompositionPrinciple . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3 Floquet theory 39
3.1 Fourieranalysison abeliangroups . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2 Periodicmanifoldsand vectorbundles . . . . . . . . . . . . . . . . . . . . . . 39
3.3 Floquet decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.4 PeriodicLaplacian on amanifold . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.5 Harmonicextension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.6 Periodiccoverings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4 Construction ofa periodicmanifold 51
4.1 Metricestimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.2 Constructionoftheperiodcell . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.3 Convergence oftheeigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.4 Estimateon thecylindricalends . . . . . . . . . . . . . . . . . . . . . . . . . 56
5 Periodicmanifoldjoinedby cylinders 59
5.1 Constructionoftheperiodcell . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.2 Convergence oftheeigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.3 Estimateoftheharmonicextension . . . . . . . . . . . . . . . . . . . . . . . . 62
1
Contents
6 Conformal deformation 65
6.1 Conformaldeformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
6.2 Lowerboundsfortheeigenvalues . . . . . . . . . . . . . . . . . . . . . . . . 67
6.3 Upperboundsfortheeigenvalues. . . . . . . . . . . . . . . . . . . . . . . . . 69
7 The two-dimensional case 71
7.1 Whatis differentinthetwo-dimensionalcase? . . . . . . . . . . . . . . . . . . 71
7.2 Limitform intwodimensions . . . . . . . . . . . . . . . . . . . . . . . . . . 72
7.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
7.4 Mid-degreeforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
8 Eigenvalues inspectral gaps 81
8.1 Approximatingproblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
8.2 Eigenvaluecountingfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . 87
8.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
Bibliography 93
Zusammenfassung 97
2
Introduction
We investigatespectral properties of the Laplace operator on a class of non-compact Rieman-
nianmanifolds. WeprovethatforagivennumberN wecanconstructaperiodicmanifoldsuch
that the essential spectrum of the corresponding Laplacian has at least N open gaps. Further-
more, by perturbing the periodic metric of the manifold locally we can prove the existence of
eigenvaluesinagapoftheessentialspectrum.
Gaps in the spectrum
In our context a periodic Riemannian manifold Mper is a non-compact d-dimensional Rie-
mannian manifold (d 2) with a properly discontinuous isometricaction of an abelian group
G of infinite order such that the orbit space Mper G is compact. As in the case of periodic
(cid:21)
Schro¨dingeroperatorsonecanapplyFloquettheory=toshowthatthespectrumoftheLaplacian
D onMper actingon p-forms(seeDefinition2.3.1)hasbandstructure,i.e.,thespectrum
ApTMper
specD is the locally finite union of compact intervals B D , called bands (see
ApTMper k ApTMper
[Don81] if G is abelian, [BS92] or [Gru98] for certain non-abeli(an groups)G or [RS78] in the
Schro¨dingeroperatorcase).
Here,werestrictourselvestotheLaplacianonfunctions,i.e.,wesuppose p 0. However,
via the Hodge -operator one can show that the spectrum of the Laplacian on f=unctions is the
sameasthespectrumoftheLaplacianond-forms. Furthermore,supersymmetryindimension
(cid:3)
2 allows us to show that the spectrum of D is the same for p 0, p 1 and p 2.
ApTMper
Therefore, all our results for the spectrum of the Laplacian on functio=ns rema=in true in t=hese
specialcases (seeTheorems2.3.8and 3.4.6and Corollaries 2.3.10and3.4.8).
In general, an infinite number of bands B D will overlap as in the case of the
k ApTMper
Laplacian D (cid:229) d ¶ 2 on d. Here, thespe(ctrumis )0 ¥ .
d i 1 i
OurfirstRaim=istoc=onstructRclassesof(non-compact)p[e;rio[dicmanifoldsMper withgapsin
(cid:0)
the essential spectrum of the Laplacian D on Mper acting on functions, i.e., we prove the
Mper
existenceofnon-voidintervals a b with
] ; [
specD a b 0/ ( )
Mper
] ; [= :
\ (cid:3)
Toexcludetrivialcaseswesupposethata infessspecD . Notethatfor(abelian-)periodic
Mper
manifoldsMper wealwayshaveinfessspe>cD 0.
Mper
We prove the existence of gaps in two differe=nt ways. In both cases the main idea is to
analyse a family of periodic manifolds Meper e such that Meper decouples in some sense as
e 0. By decouplingwemean that thej(unctio)nbetween twoperiod cells (seeSection 3.2)is
geometricallysmall.
!
3
Introduction
Ae
e
Xe Xe
X Me
Mper
e
Figure0.1: Construction ofaperiodic manifoldinCaseA.
CaseA:Westartwithacompactd-dimensionalRiemannianmanifoldX (withoutboundary
forsimplicity). IfG wegluetogether copiesofX modifiedintheneighbourhoodoftwo
distinct points in su=chZa way that we havZe two small cylindrical ends. The boundary of the
modified manifold Me is a d 1 -dimensional sphere of radius e 0 (see Figure 0.1). The
resulting manifold Mper is (-perio)dic. Note that Mper still depends>on e . By Floquet theory,
e (cid:0) e
theanalysisofthespectrumZofD isreducedtotheanalysisofthespectrumoftheLaplacian
Mper
e
on a period cell Me with q -periodic boundary conditions where q ˆ (see Section 3.4). The
dual group Gˆ ˆ 1 is usually identified with 0 2p (see SectionZ3.1). Here, a period cell
2
Me isaclosed=suZbs(cid:24)=etSoftheperiodicmanifoldMepe[r ;such[ thatMeper istheunionofalltranslates
of Me and such that MÆe does not intersect any other translate of Me . Note that the spectrum
of D q is discrete. We denote the eigenvalues written in increasing order by l q Me counting
Me k
multiplicities. In thesameway,let l q X denotethespectrumoftheLaplacian(D )on X.
k X
We provethefollowing(seeTheor(em)4.3.1and Corollary4.3.2):
Theorem. Theq -periodiceigenvaluesl q Me convergeuniformlyinq ˆ 1totheeigen-
k
valuel X ase 0foreveryk . Inp(artic)ular,ifthek-thandthe k Z1=-sSteigenvalueof
k 2 (cid:24)
the Lapla(ci)an D on X satisfyl XN l X , then there is a gap be(tw+een)the k-th and the
X! k2 k 1
k 1 -stbandofD , i.e., ( )< + ( )
Mper
( + ) e
B D B D 0/
k Mper k 1 Mper
( e ) ( e )= ;
+
providede is smallenough. \
Note that the convergence of the eigenvalue l q Me is not uniform in k (see page 8).
k
Therefore we can prove that an arbitrary finite numb(er o)f gaps occur if e is small enough.
We can extend the theorem to the case of an arbitrary finitely generated abelian group G (see
Figure0.2). WecanalsoadmitlongthincylindersoffixedlengthL 0betweenthecylindrical
ends as in Figure 0.3: TheLaplacian of theresultingperiodicmani>foldMper stillhas gapsif e
e
issmallenough(cf. Theorem5.2.1and Corrollary5.2.2). Thisresultwasoriginallymotivated
by workofC. Anne´ (see[Ann87]and [Ann99]).
Case B: In the second class of examples, we start with a G -periodicReiemannian manifold
Mper (for simplicity) without boundary. We perturb the metric gper of Mper conformally by a
factor r 2, i.e., we set gper : r 2gper and denote the resulting Riemannian manifold by Mper.
e e e e
Here r e is a family of stri=ctly positivesmooth periodic functions on Mper converging point-
wise t(o th)e indicator function of a set Xper. We suppose that Xper is the disjoint union of the
4
Introduction
Mper
e
Figure0.2: Amanifoldperiodicwithrespecttoagroupgeneratedbytwoelements(like 2or ).
p
Z Z(cid:2)Z
Ce
Mper
Me L Me e
Figure 0.3: A periodic manifold with long thin cylinders obtained by taking Me andCe asnew period
cellMee . e e
translatesofaclosedsubsetX ofMper suchthatthereexistsaperiodcellM withX MÆ. Sup-
posefurtherthatnormalcoordinateswithrespectto¶ X aredefinedonM X (seeSection6.1).
(cid:26)
Thisconditionrestricts thegeometry ofX. For example,a centered sphere in acube as period
n
cellsatisfiesthiscondition. DenotebyMe themanifoldM withmetricgpeer (seeFigure0.4and
Figure 0.5). Our second result is the following (see Theorem 6.1.2 and Corollary 6.1.3, for
theDefinitionoftheNeumannLaplacian D N seeDefinition2.3.3):
X
Theorem. SupposethatMper isofdimensiond 3. Then theq -periodiceigenvaluesl q Me
k
converge uniformly in q Gˆ to the eigenvalue l N X of the Neumann Laplacian on (X a)s
(cid:21) k
e 0, for every k . In particular,if the k-th and(the) k 1 -st eigenvalue of the Neumann
2
Laplacian D N on X sNatisfy l N X l N X , then ther(e +is a)gap between the k-th and the
! X 2 k k 1
k 1 -stbandofD provide(de)i<s sm+all(en)ough.
Mper
( + ) e
The two-dimensional case has to be treated separately. In this case we only prove that at
leastan arbitrary finitenumberofgapsexistsifMper isacylinder 1, seeChapter7.
R S
The proof of the preceding two theorems is based on the Min-Max Principle (see Theo-
(cid:2)
rem1.3.3). Themaindifficultyherecomesfromthefact thatnotonlythequadraticform(cor-
responding to the Laplacian on Me ) but also the L -norm on Me depends on e . We therefore
2
compare the Rayleigh quotients for parameter-dependent Hilbert spaces (see Theorem 1.4.2).
This idea is motivated by [Fuk87] and [Ann87], but we prove a slightly different version.
One importantingredientin provingthepreceding two theorems is a boundof theL -norm of
2
eigenfunctionsof D q on the cylindricalends (Case A)resp. on M X (Case B) convergingto
Me
0 as e 0 (seeTheorem 4.4.1resp. Theorem 6.2.1, theestimatesused there are motivatedby
n
[Ann94]).
!
In both cases gaps occur when there is a period cell Me such that a neighbourhood of the
boundary of Me is small in some sense. Note that in Case A and Case B the volume of the
e -depending part Ae (resp. Ae Ce in the case of Meper) and Me X converges to 0. It seems
that it is important to have a mechanism which “separates” or “decouples” in some sense the
[ n
differenttranslatesofa periodcell.
e
5
Introduction
Xper
M
X
Mper
Figure0.4: InCaseBthe 2-periodic manifold Mper isgiven. Wechoose aperiodcellM suchthatthe
Z
periodic subset Xper does notintersect theboundary ofM. Wefurther suppose that normal coordinates
withrespect to¶ X aredefinedonM X.
n
Mper
e
Figure0.5: An,alasimperfect,attempttopicturetheconformallyperturbedmanifoldMperobtainedby
e
scalingthemanifoldMper ofFigure0.4outsidethegreyareaXper.
Eigenvalues in gaps
Asanapplicationofourresultsonspectralgaps,weperturbthemetricoftheperiodicmanifold
Mper locally and obtain eigenvalues in a gap of the periodic Laplacian. Suppose that Mper
Mper is one of the periodic manifolds with period cell M constructed before with metric gp=er
e
such that ( ) holds. Since we will apply regularity theory we suppose that ¶ M is smooth (see
e.g. the periodic manifold in Figure 0.2). Let l a b . If l is too close to a or b we possibly
(cid:3)
have to choose a smaller e (see Corollary 8.1.2).]S;up[pose further that r t is a family
2 t 0
ofstrictlypositivesmoothfunctionson Mper such thatr 0 1 and such( th(at))r (cid:21)t isequal to
1 outside a compact set K and such that r t is equal t(o t)=1 on a compact se(t K) KÆ for
1 2 1
all t 0. Suppose further that K and K h(av)e (piecewise)+smooth boundary and non-empty
1 2 (cid:26)
interior. We denote by M t (resp. K t and K t ) the manifold Mper (resp. K and K )
(cid:21) 1 2 1 2
with metric g t r t 2g(per)conformal(to)gper. Ro(ug)ly speaking, we blow up thearea K . In
1
particular, the(ar)e=a K(is)scaled bythefactort 1 1(see Figure0.6).
2
+
(cid:21)
6
