Table Of ContentProgress in Mathematics
Volume246
SeriesEditors
HymanBass
JosephOesterle´
AlanWeinstein
Sorin Dragomir
Giuseppe Tomassini
Differential Geometry and
Analysis on CR Manifolds
Birkha¨user
Boston • Basel • Berlin
GiuseppeTomassini
SorinDragomir
ScuolaNormaleSuperiore
Universita`degliStudidellaBasilicata
ClassediScienze
DipartimentodiMatematica
PiazzadeiCavalieri7
ContradaMacchiaRomana
65126Pisa
85100Potenza
Italy
Italy
MathematicsSubjectClassification(2000):32T15,32V05,32V15,32V20,35B65,35D05,35D10,
53C07,53C10,53C12,53C15,53C17,53C21,53C25,53C26,53C40,53C43,53C50,53D10
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Contents
Preface .......................................................... ix
1 CRManifolds ................................................ 1
1.1 CRmanifolds................................................ 3
1.1.1 CRstructures ......................................... 3
1.1.2 TheLeviform......................................... 5
1.1.3 CharacteristicdirectionsonnondegenerateCRmanifolds..... 8
1.1.4 CRgeometryandcontactRiemanniangeometry ............ 10
1.1.5 TheHeisenberggroup .................................. 11
1.1.6 EmbeddableCRmanifolds .............................. 14
1.1.7 CRLiealgebrasandCRLiegroups....................... 18
1.1.8 TwistorCRmanifolds .................................. 21
1.2 TheTanaka–Websterconnection................................ 25
1.3 Localcomputations........................................... 31
1.3.1 Christoffelsymbols .................................... 32
1.3.2 Thepseudo-Hermitiantorsion............................ 36
1.3.3 Thevolumeform ...................................... 43
1.4 Thecurvaturetheory.......................................... 46
1.4.1 Pseudo-HermitianRicciandscalarcurvature ............... 50
1.4.2 Thecurvatureforms(cid:2)βα................................. 51
1.4.3 Pseudo-Hermitiansectionalcurvature ..................... 58
1.5 TheCherntensorfield......................................... 60
1.6 CRstructuresasG-structures .................................. 68
1.6.1 Integrability........................................... 70
1.6.2 Nondegeneracy........................................ 72
1.7 ThetangentialCauchy–Riemanncomplex........................ 73
1.7.1 ThetangentialCauchy–Riemanncomplex ................. 73
1.7.2 CR-holomorphicbundles................................ 82
1.7.3 TheFro¨licherspectralsequence .......................... 84
1.7.4 Alongexactsequence .................................. 94
1.7.5 Bottobstructions....................................... 96
vi Contents
1.7.6 TheKohn–RossiLaplacian .............................. 99
1.8 ThegroupofCRautomorphisms................................106
2 TheFeffermanMetric ......................................... 109
2.1 Thesub-Laplacian............................................111
2.2 Thecanonicalbundle .........................................119
2.3 TheFeffermanmetric .........................................122
2.4 ACRinvariant...............................................135
2.5 Thewaveoperator............................................140
2.6 CurvatureofFefferman’smetric ................................141
2.7 Pontryaginforms.............................................142
2.8 Theextrinsicapproach ........................................147
2.8.1 TheMonge–Ampe`reequation............................147
2.8.2 TheFeffermanmetric...................................150
2.8.3 Obstructionstoglobalembeddability......................151
3 TheCRYamabeProblem ...................................... 157
3.1 TheCayleytransform .........................................161
3.2 Normalcoordinates...........................................165
3.3 ASobolev-typelemma ........................................176
3.4 Embeddingresults............................................193
3.5 Regularityresults.............................................196
3.6 Existenceofextremals ........................................200
3.7 Uniquenessandopenproblems .................................205
3.8 Theweakmaximumprinciplefor(cid:5) ............................207
b
4 PseudoharmonicMaps......................................... 211
4.1 CRandpseudoharmonicmaps..................................212
4.2 Ageometricinterpretation .....................................215
4.3 Thevariationalapproach ......................................218
4.4 Ho¨rmandersystemsandharmonicity ............................231
4.4.1 Ho¨rmandersystems ....................................233
4.4.2 Subellipticharmonicmorphisms .........................236
4.4.3 TherelationshiptohyperbolicPDEs ......................239
4.4.4 WeakharmonicmapsfromC(H )........................242
n
4.5 Generalizationsofpseudoharmonicity ...........................246
4.5.1 Thefirstvariationformula...............................248
4.5.2 Pseudoharmonicmorphisms .............................252
4.5.3 Thegeometricinterpretationof F-pseudoharmonicity .......255
4.5.4 Weaksubelliptic F-harmonicmaps .......................256
5 Pseudo-EinsteinianManifolds................................... 275
5.1 Thelocalproblem ............................................275
5.2 Thedivergenceformula .......................................279
5.3 CR-pluriharmonicfunctions....................................280
Contents vii
5.4 Morelocaltheory ............................................294
5.5 Topologicalobstructions.......................................296
5.5.1 ThefirstChernclassofT1,0(M)..........................296
5.5.2 ThetracelessRiccitensor ...............................299
5.5.3 TheLeeclass .........................................301
5.6 Theglobalproblem...........................................304
5.7 TheLeeconjecture ...........................................312
5.7.1 QuotientsoftheHeisenberggroupbyproperlydiscontinuous
groupsofCRautomorphisms ............................313
5.7.2 RegularstrictlypseudoconvexCRmanifolds ...............318
5.7.3 TheBocksteinsequence ................................320
5.7.4 Thetangentspherebundle...............................321
5.8 Pseudo-Hermitianholonomy ...................................329
5.9 QuaternionicSasakianmanifolds ...............................331
5.10 Homogeneouspseudo-Einsteinianmanifolds......................341
6 Pseudo-HermitianImmersions .................................. 345
6.1 ThetheoremofH.Jacobowitz..................................347
6.2 Thesecondfundamentalform ..................................351
6.3 CRimmersionsintoHn+k .....................................356
6.4 Pseudo-Einsteinianstructures ..................................361
6.4.1 CR-pluriharmonicfunctionsandtheLeeclass ..............361
6.4.2 Consequencesoftheembeddingequations .................364
6.4.3 ThefirstChernclassofthenormalbundle .................369
7 QuasiconformalMappings...................................... 377
7.1 Thecomplexdilatation........................................377
7.2 K-quasiconformalmaps.......................................384
7.3 ThetangentialBeltramiequations...............................385
7.3.1 ContacttransformationsofH ...........................388
n
7.3.2 ThetangentialBeltramiequationonH ...................394
1
7.4 Symplectomorphisms .........................................398
7.4.1 Fefferman’s formula and boundary behavior of
symplectomorphisms ...................................398
7.4.2 DilatationofsymplectomorphismsandtheBeltramiequations 401
7.4.3 BoundaryvaluesofsolutionstotheBeltramisystem.........403
7.4.4 AtheoremofP.Libermann ..............................403
7.4.5 Extensionsofcontactdeformations .......................405
8 Yang–MillsFieldsonCRManifolds.............................. 407
8.1 CanonicalS-connections ......................................407
8.2 InhomogeneousYang–Millsequations...........................410
8.3 Applications.................................................412
8.3.1 Triviallinebundles.....................................414
8.3.2 Locallytriviallinebundles ..............................414
viii Contents
8.3.3 Canonicalbundles .....................................416
8.4 Variousdifferentialoperators...................................418
8.5 CurvatureofS-connections ....................................421
9 SpectralGeometry ............................................ 423
9.1 Commutationformulas........................................423
9.2 Alowerboundforλ .........................................427
1
9.2.1 ABochner-typeformula ................................430
9.2.2 Twointegralidentities ..................................431
9.2.3 A.Greenleaf’stheorem .................................434
9.2.4 AlowerboundonthefirsteigenvalueofaFolland–Stein
operator ..............................................439
9.2.5 Z.JiaqingandY.Hongcang’stheoremonCRmanifolds......441
AParametrixfor(cid:2) ............................................... 445
b
References ....................................................... 463
Index............................................................ 483
Preface
∞
A CR manifold is aC differentiable manifold endowed with a complex subbundle
T1,0(M)ofthecomplexifiedtangentbundleT(M)⊗CsatisfyingT1,0(M)∩T1,0(M)=
(0)andtheFrobenius(formal)integrabilityproperty
(cid:2) (cid:3)
(cid:7)∞(T1,0(M)), (cid:7)∞(T1,0(M)) ⊆(cid:7)∞(T1,0(M)).
The bundle T1,0(M) is the CR structure of M, and C∞ maps f : M → N of CR
manifoldspreservingtheCRstructures(i.e., f∗T1,0(M)⊆T1,0(N))areCRmaps.CR
manifolds and CR maps form a category containing that of complex manifolds and
holomorphicmaps.ThemostinterestingexamplesofCRmanifoldsappear,however,
asrealsubmanifoldsofsomecomplexmanifold.Forinstance,anyrealhypersurface
M in Cn admits a CR structure, naturally induced by the complex structure of the
ambientspace
T1,0(M)=T1,0(Cn)∩[T(M)⊗C].
Let(z1,...,zn)bethenaturalcomplexcoordinatesonCn.Locally,inaneighborhood
ofeachpointof M,onemayproduceaframe{Lα :1≤α ≤n−1}ofT1,0(M).Geo-
metricallyspeaking,each Lα isa(complex)vectorfieldtangentto M.Fromthepoint
of view of the theory of PDEs, the Lα’s are purely tangential first-order differential
operators
(cid:4)n ∂
Lα = aαj(z)∂zj, 1≤α ≤n−1,
j=1
andT1,0(M)maybethoughtofasabundle-theoreticrecastingofthefirst-orderPDE
∞
systemwithcomplex-valuedC coefficients
Lαu(z)=0, 1≤α ≤n−1,
calledthetangentialCauchy–Riemannequations.Thesemaybeequallythoughtofas
beinginducedon M bytheCauchy–RiemannequationsinCn.CRfunctionsaresolu-
tionsu(z)tothetangentialCauchy–Riemannequations,andanyholomorphicfunction
definedonaneighborhoodof M willrestricttoaCRfunctionon M.
x Preface
These introductory remarks lead to two fundamental problems in CR geometry
and analysis. Given an (abstract) CR manifold, is it possible to realize it as a CR
submanifoldofCn (ofsomecomplexmanifold)?Thisisknownastheembeddability
problem,introducedtomathematicalpracticebyJ.J.Kohn[246].Thesecondproblem
iswhetheragivenCRfunctionu : M →Cextendstoaholomorphicfunctiondefined
on some neighborhood of M (the CR extension problem). Both these problems have
local and global aspects, present many intricacies, and involve scientific knowledge
frommanymathematicalfields.Thesolutiontothelocalembeddabilityproblemisdue
toA.AndreottiandC.D.Hill[13]intherealanalyticcategory.Partialsolutionsinthe
∞
C categoryareduetoL.BoutetdeMonvel[77],M.Kuranishi[263],andT.Akahori
[2]. As to the CR extension problem, it is the object of intense investigation, cf. the
monographsbyA.Boggess[70]andM.S.Baouendi,P.Ebenfelt,andL.P.Rothschild
[31]foranaccountofthepresentscientificachievementsinthisdirection.
ItshouldbecomeclearfromthisdiscussionthatCRmanifoldsandtheirstudylie
at the intersection of three main mathematical disciplines: the theory of partial dif-
ferential equations, complex analysis in several variables, and differential geometry.
WhiletheanalysisandPDEaspectsseemtohavecapturedmostoftheinterestwithin
the mathematical community, there has been, over the last ten or fifteen years, some
efforttounderstandthedifferential-geometricsideofthesubjectaswell.Itistruethat
A. Bejancu’s discovery [55] of CR submanifolds signaled the start of a large num-
ber of investigations in differential geometry, best illustrated by the monographs by
K.YanoandM.Kon[446],A.Bejancu[56],andS.DragomirandL.Ornea[125].Here
by a CR submanifold we understand a real submanifold M of a Hermitian manifold
(X,J,g), carrying a distribution H(M) that is J-invariant (i.e., JH(M) = H(M))
and whose g-orthogonal complement is J-anti-invariant (i.e., JH(M)⊥ ⊆ T(M)⊥,
where T(M)⊥ → M is the normal bundle of M in X). The notion (of a CR sub-
manifold of a Hermitian manifold) unifies concepts such as invariant, anti-invariant,
totally real, semi-invariant, and generic submanifolds. Also, the observation (due to
D.E. Blair and B.Y. Chen [64]) that proper CR submanifolds, in the sense of A. Be-
jancu,areactuallyCRmanifoldsshowsthattheseinvestigationshavethesamecentral
object, the CR category, as defined at the beginning of this preface, or by S. Green-
field [187]. The study of CR submanifolds in Hermitian manifolds, in the sense of
A.Bejancu,hasledtothediscoveryofmanyrefineddifferential-geometricproperties
(e.g.,K.YanoandM.Kon’sclassificationofCRsubmanifoldsofacomplexprojective
space,withsemiflatnormalconnection,parallel f-structureinthenormalbundle,and
thecovariantderivativeofthesecondfundamentalformofconstantlength[445])and
willsurelydevelopfurtherwithinitsownborders.Itshouldberemarkednevertheless
that as confined to Riemannian geometry (i.e., to the theory of submanifolds in Rie-
mannianmanifolds,cf.,e.g.,[91]),theabove-mentionedstudyisperhapsinsufficiently
relatedtothe(pseudo)convexitypropertiesofsubmanifoldsincomplexmanifolds,as
understoodinanalysisinseveralcomplexvariables.Tobemoreprecise,if M isareal
hypersurfaceinCnthenthefirstandsecondfundamentalformsofthegivenimmersion
describe the way M is shaped, both intrinsically (Riemannian curvature) and extrin-
sically,yetdonotdescribeaprioritheintrinsicpropertiesof M asrelatedtoitsLevi
form.Asanextremecase, M maybeLeviflatyetwillalwaysexhibit,say,curvature
Preface xi
propertiesarisingfromitsfirstfundamentalform.Ortogiveanondegenerateexample,
theboundaryoftheSiegeldomain(cid:2) ={z =(z(cid:10),z )∈Cn−1×C:Im(z )>(cid:12)z(cid:10)(cid:12)2}
n n n
(i.e.,theHeisenberggroup)admitsacontactformθ withapositivedefiniteLeviform
Gθ,andhencegθ =πHGθ +θ ⊗θ (theWebstermetric)isaRiemannianmetric,yet
noneofthemetrics gλθ, λ ∈ C∞(M), λ > 0,coincideswiththemetricinducedon
M bytheflatKa¨hlermetricofCn.
Centraltothepresentmonographisthediscovery,around1977–78,ofacanonical
linearconnection∇ oneachnondegenerateCRmanifold M ofhypersurfacetype(the
Tanaka–Webster connection) due to independent investigations by N. Tanaka [398]
and S. Webster [422]. ∇ parallelizes both the Levi form and the complex structure
oftheLevi,ormaximallycomplex,distributionof M,resemblesboththeLevi-Civita
connectionofaRiemannianmanifold,andtheChernconnectionofaHermitianman-
ifold,andisafoundationaltoolforthepseudo-Hermitiangeometryofa(nondegener-
ate)CRmanifold,whichisthemainsubjectofthisbook.Nowthecurvatureproper-
tiesof∇ areindeedtiedtotheCRstructure:forinstance,theCherncurvaturetensor
Cβαλσ,aCRinvariantofM,iscomputableintermsofthecurvatureof∇(anditscon-
tractions,suchasthepseudo-HermitianRiccitensorandthepseudo-Hermitianscalar
curvature)andCβαλσ = 0ifandonlyif M islocallyCRequivalenttothestandard
sphere in Cn+1, n > 1 (cf. S.S. Chern and J. Moser [99]). Variants of the Tanaka–
Webster connection are known already in different contexts, e.g., on CR manifolds
of higher CR codimension (R. Mizner [312]) or on contact Riemannian manifolds
(S.Tanno[401]),whosealmostCRstructureisnotintegrable,ingeneral.
After a detailed exposition of the basic facts of pseudo-Hermitian geometry of
nondegenerateCRmanifoldsinChapter1,thepresentmonographintroducesthemain
geometricobject,theFeffermanmetric,bothatoolandobjectofinvestigationofthe
first magnitude. It is due to C. Fefferman [138], who first devised it as a (Lorentz)
metricon(∂(cid:2))× S1,foragivenstrictlypseudoconvexdomain(cid:2) ⊂ Cn,inconnec-
tion with the boundary behavior of the Bergman kernel of (cid:2) and the solution to the
Dirichletproblemforthe(inhomogeneous)complexMonge–Ampe´reequation
⎧ (cid:9) (cid:10)
⎪⎨(−1)n+1det u ∂u/∂zk =1 in(cid:2),
∂u/∂zj ∂2u/∂zj∂zk
⎪⎩
u =0 on∂(cid:2)
(the existence, uniqueness, and regularity of the solution are due to S.Y. Cheng and
S.T. Yau [97]). See Chapter 2 of this book. By the mathematical creation of F. Far-
ris [137], and J.M. Lee [271], an intrinsic description of the Fefferman metric (as a
Lorentzmetricon
(cid:11) (cid:12)(cid:13)
C(M)= (cid:12)n+1,0(M)\{0} R+,
where n is the CR dimension) is available. Also, the work of G. Sparling [377],
C.R.Graham[182],L.Koch[242]–[244],helpedclarifyanumberofgeometricfacts
(e.g.,howtheFeffermanmetricmaybesingledout,intermsofcurvatureproperties,
fromthesetofallLorentzmetricsonC(M)(cf.[182]),orprovidingasimpleproof
