Characteristic foliations on maximally real submanifolds of C^n and envelopes of holomorphy

1.3 MB·English

by  Joël Merker

CHARACTERISTIC FOLIATIONS ON MAXIMALLY REAL
SUBMANIFOLDS OF Cn AND ENVELOPES OF HOLOMORPHY
4
0
0
JOE¨LMERKERANDEGMONTPORTEN
2

n
a ABSTRACT. LetS beanarbitraryreal2-surface, withorwithoutboundary,contained
J inahypersurfaceM ofC2,withS andM ofclass 2,α,where0 < α < 1. IfS is
4 totallyrealexceptatfinitelymanycomplextangenciesCwhicharehyperbolicinthesense
1 ofE.Bishopandiftheunionofseparatricesisatreeofcurveswithoutcycles,weshow
thateverycompactK ofS isCR-, -andLp-removable(Theorem1.3). Ourpurely
] localtechniquesenableustoformulaWtesubstantialgeneralizations ofthisstatement,for
V theremovabilityofclosedsetsintotallyreal1-codimensionalsubmanifoldscontainedin
genericsubmanifoldsofCRdimension1.
C
.
h
t
a Tableofcontents
m
PartI .............................................................................................1.
1.Introduction ...................................................................................1.
[
2.DescriptionoftheproofofTheorem1.2andorganizationofthepaper............................8.
3.StrategyperabsurdumfortheproofsofTheorems1.2’and1.4 .................................13.
1
4.Constructionofasemi-localhalf-wedge ........................................................19.
v 5.ChoiceofaspecialpointofCnrtoberemovedlocally ..........................................32.
2 PartII ...........................................................................................45.
4 6.ThreepreparatorylemmasonHo¨lderspaces ...................................................45.
7.Familiesofanalyticdiscshalf-attachedtomaximallyrealsubmanifolds..........................47.
1
8.Geometricpropertiesoffamiliesofhalf-attachedanalyticdiscs..................................55.
1 9.EndofproofofTheorem1.2’:applicationofthecontinuityprinciple ............................63.
0 10.ThreeproofsofTheorem1.4 .................................................................69.
4 1121..PWro-orefsmoofvTabhieloitryemim1p.l1ieasnLdpo-rfeTmhoevoarebmilit1y.3.........................................................................................................7738..
0
13.Applicationstotheedgeofthewedgetheorem.................................................87.
/ 14.Anexampleofanonremovablethree-dimensionaltorus .......................................91.
h
15.References...................................................................................96.
at [With 24 figures]
m
:
v 1.INTRODUCTION
§
i
X Inthepastfifteenyears,remarkableprogresshasbeenmadetowardstheunderstand-
r ing of the holomorphic extendability properties of CR functions. At the origin of this
a
development,themostfundamentalachievementwasthedeepdiscovery,duetotheeffort
of numerous mathematicians, that the so-called CRorbitsare the adequate underlying
objectsforthesemi-localCRanalysisonageneralembeddedCRmanifold. Asaninde-
pendentandnowestablishedtheoryinseveralcomplexvariables,onemayfindaprecise
correspondencebetweensuchorbitsandprogressivelyattachedanalyticdiscscoveringa
thick partof the envelopeof holomorphyof CR manifolds, cf. [B], [Trv], [Tr1], [Tu1],
[BER],[Tu2],[M1],[J2]and[P3]forarecentsynthesis.
Withinthisframework,itbecamemathematicallyaccessibletoendeavourthegeneral
study of removablesingularitieson embeddedCR manifoldsM Cn of arbitraryCR
⊂
Date:2008-2-1.
2000Mathematics SubjectClassification. Primary: 32D20. Secondary: 32A20, 32D10, 32V10, 32V25,
32V35.
1
2 JOE¨LMERKERANDEGMONTPORTEN
dimensionandofarbitrarycodimension,notnecessarilybeingtheboundariesof(strictly)
pseudoconvex domains. With respect to their size or “mass”, the interesting singular-
ities can be essentially ordered by their codimension in M. For instance, provided it
doesnotperturbthefactthatM consistsofasingleCRorbit,anarbitraryclosedsubset
C M whichisofvanishing2-codimensionalHausdorffcontentisalways removable,
⊂
asisshownin[CS]inthehypersurfacecaseandin[MP3],Theorem1.1,inarbitrarycodi-
mension. Hence one is left to study the removabilityof singularitiesof codimensionat
mosttwo. Sincethegeneralproblemofcharacterizingremovabilityseemsatthemoment
tobeoutofreach(evenforM beingahypersurface),itisadvisibletofocusongeometri-
callyaccessiblesingularities,namelysingularitiescontainedinaCRsubmanifoldofM.
Acompletestudyoftheautomaticremovabilityoftwo-codimensionalsingularitiesmay
be found in Theorem 4 of [MP1]. Having in mind the classical Painleve´ problem, we
will mainly consider in this paper singularities which are closed sets C contained in a
codimensionone submanifoldM1ofM whichisgeneric inCn.
The known results on singularities of codimension one can be subdivided into two
stronglydifferentgroupsaccordingtotheCR dimensionofM. IfCRdimM 2,then
≥
agenerichypersurfaceM1 M isitselfaCRmanifoldofpositiveCRdimension,and
⊂
singularitiesC M1canbeunderstoodonthebasisoftheinterplaybetweenC andthe
⊂
CRorbitsofM1.DeepresultsinthisdirectionwereestablishedwhenMisahypersurface
ofCnin[J4],[J5]andthengeneralizedtoCRmanifoldsofarbitrarycodimensionin[P1]:
thegeometricconditioninsuringautomaticremovabilityissimplythatCdoesnotcontain
anyCRorbitofM1.
On the other hand, if CRdimM = 1 the geometricsituation becomeshighlydiffer-
ent, as a generichypersurfaceM1 M is now (maximal) totally real. Fortunately,as
⊂
a substitutefortheCRorbitsofM1, onecanconsidertheso-calledcharacteristicfolia-
tion of M1, obtainedby integratingthe characteristic line field TcM TM1. But
M1
removabilitytheoremsexploitingthisconceptwereonlyknownforhy|pers∩urfacesinC2
and,untilveryrecently,onlyinthestrictlypseudoconvexcase. Furthermore,ageometric
conditioninsuringautomaticremovablitityhasnotyetbeenclearlydelineated.
Hence, with respect to the current state of the art, there was a two-fold gap about
codimensiononeremovablesingularitiescontainedingenericsubmanifoldsM ofCRdi-
mensionone:firstly,toestablishasatisfyingtheoryfornon-pseudoconvexhypersurfaces
inC2andsecondly,tounderstandthesituationinhighercodimension.Thissecondmain
task wasformulatedasthe firstopenproblemin a listp.432of[J5](see alsothe com-
mentspp.431–432abouttherelativegeometricsimplicityofthecase CRdimM 2).
≥
Apriori,itisnotclearatallwhetherthetwodirectionsofresearcharerelatedsomehow,
but in the present work, we shall fill in this two-fold gap by devising a new semi-local
approachwhichappliesuniformlywithrespecttocodimension.
Forthedetaileddiscussionofourresultwehavetointroducesometerminologywhich
willbeusedthroughoutthearticle. LetM beagenericsubmanifoldofCn andletC be
aclosedsubsetofM. Recallfrom[MP3]thatawedgelikedomainattachedtoageneric
submanifoldM CnisadomaincontainingalocalwedgeofedgeM ateverypointof
′ ′
⊂
M . Ourwedgelikedomainswillalwaysbenonempty. Letusdefinethreebasicnotions
′
of removability. Firstly, we say that C isCR-removableif there exists a wedgelike do-
main attachedtoM towhicheverycontinuousCRfunctionf 0 (M C)extends
W ∈CCR \
holomorphically. Secondly, as in [MP3], p. 486, we say that C is -removableif for
W
everywedgelikedomain attachedtoM C,thereisawedgelikedomain attached
1 2
W \ W
to M and a wedgelike domain attached to M C such that for every
3 1 2
W ⊂ W ∩W \
CHARACTERISTICFOLIATIONSANDENVELOPESOFHOLOMORPHY 3
holomorphic function f ( ), there exists a holomorphic function F ( )
1 2
whichcoincideswithf in∈ O.WThirdly,withp R + satisfyingp ∈1,OweWsay
3
thatC isLp-removableifeWverylocallyintegrabl∈efun∪cti{on∞f } Lp (M)wh≥ichisCRin
∈ loc
thedistributionalsenseonM C isinfactCRonallofM.
\
Thefirstnotionofremovabilityisageneralizationofthekindofremovabilityconsid-
ered in most of the pioneeringpapers[CS], [D], [FS], [J1], [KR], [L], [Lu], [St] about
removablesingularitiesinboundariesofdomainsD Cn. Weobservethatawedge-
⊂⊂
like opensetattachedto a hypersurfaceM is justa (global)one-sidedneighborhoodof
M, namely a domain ω with ω M such that for every point p M, the domain ω
contains the intersection of a nei⊃ghborhood of p in Cn with one s∈ide of M. If now a
closedsetC containedina 1-smoothboundedboundary∂D isCR-removable,thenan
C
application of the Hartogs-Bochnertheorem shows thatCR functionson ∂D C can be
\
holomorphicallyextendedtoD. Thesecondnotionofremovabilityisawaytoisolatethe
partofthequestionrelatedtoenvelopesofholomorphy.Thethirdnotionofremovability
hastheadvantageofbeingcompletelyintrinsicwithrespecttoM andmayberelevantin
thestudyofnon-embeddableCRmanifolds.
To avoid confusion, we state precisely our submanifold notion: Y is a submanifold
ofX if Y andX are equippedwith a manifoldstructure,if thereexistsanimmersioni
ofY intoX andifthemanifoldtopologyofY andthetopologyofi(Y)inheritedfrom
thetopologyofX coincide,sothatonemayidentifythesubmanifoldY withthesubset
i(Y) X. Furthermore,oursubmanifoldswillalwaysbeconnected.
⊂
Let us now enter the discussion of the case n = 2. Here we shall denote the sub-
manifold M1 M, which is a surface in C2, by S. In [B], E. Bishop showed that
a two-dimensio⊂nal surface in C2 of class at least 2 having an isolated complex tan-
C
gency at one of its points p may be represented by a complex equation of the form
w =zz¯+λ(z2+z¯2)+o(z 2),intermsoflocalholomorphiccoordinates(z,w)vanishing
| |
at p, wherethe realparameterλ [0, ) isa biholomorphicinvariantofS. The point
∈ ∞
p is said to beellipticif λ [0,1), parabolicif λ = 1 andhyperbolicif λ (1, ).
∈ 2 2 ∈ 2 ∞
Recall that M is called globallyminimalif it consists of a single CR orbit (cf. [Tr1],
[Tr2];[MP1],pp.814–815;and[J4],pp.266–269).Throughoutthispaper,weshallwork
inthe 2,α-smoothcategory,where0<α<1. Ourfirstmainnewresultisasfollows.
C
Theorem 1.1. Let M be a globally minimal 2,α-smooth hypersurface in C2 and let
C
D M bea 2,α-smoothsurfacewhichis
⊂ C
(a) 2,α-diffeomorphictotheunit2-discofR2and
C
(b) totally real outside a discrete subset of isolated complex tangencies which are
hyperbolicinthesenseofE.Bishop.
TheneverycompactsubsetK ofDisCR-, -andLp-removable.
W
Asacorollary,oneobtainsacorrespondingresultaboutholomorphicextensionfrom
∂Ω K for the case that M is the boundary of a relatively compact domain Ω C2.
\ ⊂
Notethat∂Ωisautomaticallygloballyminimal([J4],Section2). Wewillfirstrecallthe
historicalbackgroundofTheorem1.1andexplainafterwardsonthisbasisthemainideas
andtechniquesnecessaryfortheproof.
In1988,applyingaglobalversionoftheKontinuita¨tssatz,B.Jo¨ricke[J1]established
aremarkabletheorem: everycompactsubsetofatotallyreal 2-smooth2-disclyingon
the boundary of the unit ball in S3 = ∂B C2 is CR-reCmovable. This discovery
2
⊂
motivated the work [FS] by F. Forstneriˇc and E.L. Stout, where it is shown that every
2-smoothcompact2-disccontainedinastrictlypseudoconvex 2-smoothboundary∂Ω
C C
4 JOE¨LMERKERANDEGMONTPORTEN
contained in a 2-dimensional Stein manifold which is totally real except at a finite
M
number of hyperbolic complex tangencies is removable; the proof mainly relies on a
previousworkbyE.BedfordandW.Klingenbergaboutthehullsof2-spherescontained
in such strictly pseudoconvexboundariesΩ , which may be filled by Levi-flat 3-
⊂ M
spheres after a generic small perturbation ([BK], Theorem 1). Indirectly, it followed
from[J1]and[FS]thatsuchcompacttotallyreal2-discsD ∂Ω(possiblyhavingfinitely
⊂
many hyperboliccomplex tangencies) are (Ω)-convex and in particular polynomially
convexif D = B and = C2, thankstoOapreviouswork[St] byE.L.Sout, whereit
2
M
isshown(TheoremII.10)thatacompactsubsetK ofa 2-smoothstrictlypseudoconvex
C
boundary∂ΩinaSteinmanifoldisremovableifandonlyif K is (Ω)-convex.Itisalso
O
established in [FS] that a neighborhoodof an isolated hyperbolic complex tangency in
C2ispolynomiallyconvex.Thesepapershavebeenfollowedbythework[D],wherethe
questionof (Ω)-convexityofarbitrarycompactsurfaces S (withorwithoutboundary,
O
notnecessarilydiffeomorphictoa2-disc)containedina 2-smoothstrictlypseudoconvex
domainΩ C2isdealtwithdirectly.UsingK.Oka’schCaracterizationoftheenvelopeof
⊂
acompact,J.DuvalshowsthattheessentialhullK :=K Kmustcrosseveryleaf
ess (Ω)\
ofthecharacteristicfoliationonthetotallyrealpartofS anOdhededucesthatacompact
2-dischavingonlyhyperboliccomplextangenciebsis (Db)-convex.
O
All the above proofs heavily rely on strong pseudoconvexity, in contrast to the ex-
perience, familiar at least in the case CRdimM 2, that removabilityshould depend
≥
rather on the structure of CR orbits than on Levi curvature. The first theorem for the
non-pseudoconvexsituationwasestablishedbythesecondauthorin[P2]. Heprovedthat
everycompactsubsetofa totallyrealdisc embeddedin agloballyminimal -smooth
∞
hypersurface in C2 is always CR-removable. We would like to point out thCat, seeking
theoremswithoutanyassumptionofpseudoconvexityleadstosubstantialopenproblems,
becauseonelosesalmostallofthestronginterweavingsbetweenfunction-theoretictools
andgeometricargumentswhicharevalidinthepseudoconvexrealm,forinstance: Hopf
Lemma, plurisubharmonic exhaustions, envelopes of function spaces, local maximum
modulusprinciple,Steinneighborhoodbasis,etc.
To discuss the main elements of our approach, let us briefly explain the geometric
setupoftheproofofTheorem1.1. Thecharacteristicfoliationhasisolatedsingularities
at the hyperbolic points, where it looks like the phase diagram of a saddle point. In
particulartherearefourlocalseparatricesaccumulatingorthogonallyateachhyperbolic
point. Hence we can decompose the 2-disc D as a union D = T D , where T
D o D
∪
consistsoftheunionofthehyperbolicpointsofDtogetherwiththeseparatricesissuing
fromthem,andwhereD := D T istheremainingopensubmanifoldofD,contained
o D
\
in the totally realpart of D. By H. Poincare´ and I. Bendixson’stheory, T is a tree of
D
2,α-smooth curveswhich contains no subset homeomorphicto the unit circle, cf. [D].
C
Accordingly,wedecomposeK :=K C ,whereK :=K T isaproperclosed
TD∪ o TD ∩ D
subsetofthetreeT andwhereC :=K D isarelativelyclosedsubsetofD .
D o o o
∩
The hardpartof the proof,which wasactually the starting pointof the wholepaper,
willconsistinremovingtheclosedsubsetC ofthe2-dimensionalsurfaceS :=D lying
o o
inM T . ThereaftertheremovaloftheremainingpartK willbedonebymeansof
\ D TD
aninvestigationofthebehaviouroftheCRorbitsnearT ,closeinspirittoourprevious
D
methodsin[MP1](see Section12belowforthedetails).
Letusformulatethefirstcrucialpartoftheaboveargumentasanindependenttheorem
about the removalof closed subsets containedin a totally real surface S. We pointout
thatnowS mayhavearbitrarytopology.
CHARACTERISTICFOLIATIONSANDENVELOPESOFHOLOMORPHY 5
Theorem1.2. LetM bea 2,α-smoothgloballyminimalhypersurfaceinC2,letS M
C ⊂
be a 2,α-smooth surface, open or closed, with or without boundary, which is totally
C
realateverypoint. LetC beaproperclosedsubsetofS andassumethatthefollowing
topologicalconditionholds:
c C : For every closed subset C C, there exists a simple 2,α-smooth curve γ :
FS{ } ′ ⊂ C
[ 1,1] S, whose range is contained in a single leaf of the characteristic
− →
foliation c (obtainedbyintegratingthecharacteristiclinefieldTcM TS),
FS |S ∩
with γ( 1) C , γ(0) C and γ(1) C , such that C lies completely in
′ ′ ′ ′
− 6∈ ∈ 6∈
oneclosedsideofγ[ 1,1]withrespecttothetopologyofS inaneighborhood
−
ofγ[ 1,1].
−
ThenC isCR-, -andLp-removable.
W
The condition c C is a common condition on C and on the characteristic folia-
FS{ }
tion c, namely on the relative dispositionof c with respectto C, notonly on S; an
FS FS
illustrationmaybefoundin FIGURE 2 below. Inthestrictlypseudoconvexcontext,this
conditionappearedimplicitlyduringthecourseoftheproofsgivenin[D]. Notethatthe
relevanceofthecharacteristicfoliationhadearlierbeendiscoveredincontactgeometry,
cf. [Be],[E]. Itisinterestingtonoticethatitre-appearsinthesituationofTheorem1.2,
wheretheunderlyingdistributionTcM isallowedtobeveryfarfromcontact.
Asisknown,itfollowsfromasubcaseofH.Poincare´ andI.Bendixson’stheorythat
if S is diffeomorphicto a real 2-disc or if S = D as above, then c C is automat-
o FS{ }
ically satisfied for an arbitry compact subset C of S. On the contrary, it may be not
satisfiedwhenforinstanceS isanannulusequippedwitharadialfoliationtogetherwith
C containingacontinuousclosedcurvearoundtheholeofS. Crucially,itiselementary
to construct an example of such an annulus which is truly nonremovable. Indeed, the
small closed curve C which consists of the transversal intersection of a strictly convex
boundary∂D with a complex line close to a boundarypointmay be enlargedas a thin
maximallyrealstripS ∂D whichisdiffeomorphictoanannulus;inthissetting,C is
⊂
obviouslynonremovableandthecharacteristicfoliationiseverywheretransversaltoC.
Consequently,thegeometriccondition c C istheoptimaloneinsuringautomaticre-
FS{ }
movabilityforallchoicesofM,S andC. Furtherexamplesofclosedsubsetsinsurfaces
witharbitrarygenusequippedwithsuchfoliationsmaybeexhibited.
IntheproofofTheorem1.2,aftersomecontractionC ofC,wemayassumethatno
′
pointofC islocallyremovable(see Sections2and3below).Thentheveryassumption
′
c C yieldstheexistenceofacharacteristicsegmentγ[ 1,1],suchthatC liesonone
FS{ } − ′
sideofγ[ 1,1]. Reasoningbycontradiction,ouraimistoshowthatthereexistsatleast
−
onespecialpointp C γ( 1,1),whichislocally CR-, -andLp-removable. The
′
∈ ∩ − W
choiceofsuchapointp,achievedinSection5below,willbenontrivial.
The strategy for the local removalof p is to constructan analytic disc A such thata
segmentofitsboundary∂AisattachedtoS andtouchesC inonlyonepointp. Several
′
geometricalassumptionshavetobemettoensurethatasufficientlyrichfamilyofdefor-
mationsofAhaveboundariesdisjointfromC ,thatanalyticextensionalongthesediscs
′
ispossible(i.e.appropriatemomentconditionsaresatisfied),andthattheunionofthese
gooddiscsislargeenoughtogiveanalyticextensiontoaone-sidedneighborhoodofM:
thisiswherethe(semi)localizationandthechoiceofthespecialpointp C willbekey
′
∈
ingredients.Letusexplainwhylocalizationiscrucial.
6 JOE¨LMERKERANDEGMONTPORTEN
Workinggloballly,thesecondauthorproducedin[P2]aconvenientdiscbyapplying
thepowerfulE.BedfordandW.Klingenbergtheoremtoanappropriate2-spherecontain-
inganeighborhoodoftheentire singularityC . Thismethodrequiresglobalproperties
′
ofS likeS beingatotallyreal2-disc,whichensurestheexistenceofaniceSteinneigh-
borhoodbasisofC . Alreadyforrealdiscswithisolatedhyperbolicpoints,itisnotclear
′
whetherthisargumentcanbegeneralized(however,wewouldliketomentionthatrecent
resultsofM.Slaparin[Sl]indicatethatthiscouldbepossibleatleastifthegeometrynear
thehyperbolicpointssatisfiessomeadditionalassumptions). InthecasewhereM isan
arbitrarygloballyminimalhypersurface,whereShasarbitrarytopologyandhascomplex
tangencies,thereductiontoE.BedfordandW.Klingenberg’stheoremseemsimpossible,
cf. the exampleofan unknottednonfillable2-spherein C2 constructedby J.E. Fornæss
andD.Main[FM]. Also,totheauthors’knowledge,thepossibilityoffillingbyLevi-flat
3-spheresthe(notnecessarilygeneric)2-sphereslyingonanonpseudoconvex hypersur-
faceisadelicateopenproblem. Inaddition,forthehighercodimensionalgeneralization
ofTheorem1.2,theideaofglobalfillingseemstobeirrelevantatpresenttimes,because
noanalogoftheE.BedfordandW.Klingenbergtheoremisknownindimensionn 3.
≥
As we aim to deal with surfaces S having arbitrary topology and to generalize these
results in arbitrary codimension, we shall endeavour to firmly localize the removability
arguments,usingonlysmallanalyticdiscs.
Thus,ourwaytoovercometheseobstaclesistoconsiderlocaldiscsAwhichareonly
partiallyattachedtoS.Thedelicatepointisthatwehaveatthesametime(i)tocontrolthe
geometryof∂Anearp C and(ii)toguaranteethattherestoftheboundarystaysinthe
′
∈
regionwhereholomorphicextensionisalreadyknown.Infact,(ii)willbeincorporatedin
ourveryspecialandtrickychoiceofp C . For(i),wehavetosharpenknownexistence
′
∈
theorems about partially attached analytic discs and to combine it with a careful study
of the complex/realgeometryofthe pair (M,S). Importantly,ourconstructionof such
analyticdiscsisachievedelementarilyinaself-containedway. Aprecisedescriptionof
the proofin the hypersurfacecase (only)may be foundin Section 2 below. With some
substantial extra work, we shall generalize this purely local strategy of proof to higher
codimension.
Toconcludewiththeremovalofsurfaces,letusformulateamoregeneralversionof
Theorem1.1,whitouttherestrictedtopologicalassumptionthatS bediffeomorphictoa
realdisc.ApplyingTheorem1.2fortheremovalofK (S T )andaslightgeneralization
S
∩ \
ofTheorem4(ii)in[MP1]fortheremovalofK T (moreprecisionswillbegivenin
S
∩
13 below), we shall obtain the following statement, implying Theorem 1.1 as a direct
§
corollary.
Theorem1.3. LetM bea 2,α-smoothgloballyminimalhypersurfaceinC2,letS M
C ⊂
bea 2,α-smoothtotallyrealsurface,openorclosed,withorwithoutboundary,whichis
C
totallyrealoutsideadiscretesubsetofisolatedcomplextangencieswhicharehyperbolic
in the sense of E. Bishop. Let T be the union of hyperbolic points of S together with
S
allseparatricesissuedfromhyperbolicpointsandassumethatT doesnotcontainany
S
subsetwhichishomeomorphichtotheunitcircle.LetK beapropercompactsubsetofS
andassumethat c K (S T ) holds.
ThenK isCRF-,S\T-Sa{ndL∩p-re\moSva}ble.
W
Aswasalreadyemphasized,ourmainmotivationforthisworkwasto devisealocal
strategyofproofforTheorems1.1,1.2and1.3inordertogeneralizethemtohighercodi-
mension. In fact, we will realize the programsketched abovefor generic submanifolds
ofCRdimension1andofarbitrarycodimension.Thus,letM bea 2,α-smoothglobally
C
CHARACTERISTICFOLIATIONSANDENVELOPESOFHOLOMORPHY 7
minimalgenericsubmanifoldofcodimension(n 1)inCn, henceofCRdimension1,
−
wheren 2. LetM1 beamaximallyreal 2,α-smoothone-codimensionalsubmanifold
ofM wh≥ichisgenericinCn. AsinthesurfCacecase,M1carriesacharacteristicfoliation
c , whoseleavesaretheintegralcurvesofthelinedistributionTM1 TcM . Of
FM1 ∩ |M1
coursetheassumptionthatthesingularityliesononesideofsomecharacteristicsegment
is nolongerreasonable. We willgeneralizeitasaconditionrequiring(approximatively
speaking)thattherebealwaysacharacteristicsegmentwhichisaccessiblefromthecom-
plementofC inM1alongonedirectionnormaltothecharacteristicsegment.
The generalization of Theorem 1.2, which is our principal result in this paper, is as
follows.
Theorem1.2’. LetM,M1, c beasaboveandletC beaproperclosedsubsetofM.
FM1
Assumethatthefollowingtopologicalcondition,meaningthatC isnottransversaltothe
characteristicfoliation,holds:
c C : For every closed subset C C, there exists a simple 2,α-smooth curve
FM1{ } ′ ⊂ C
γ : [ 1,1] M1 whose range γ[ 1,1] is contained in a single leaf of the
− → −
characteristicfoliation c withγ( 1) C ,γ(0) C andγ(1) C ,there
FM1 − 6∈ ′ ∈ ′ 6∈ ′
exists a local(n 1)-dimensionaltransversal R1 M1 to γ passingthrough
− ⊂
γ(0)andthereexistsathinelongatedopenneighborhoodV ofγ[ 1,1]inM1
1
−
such thatif π c : V1 R1 denotesthe semi-localprojection parallelto the
FM1 →
leavesofthecharacteristicfoliation c ,thenγ(0)liesontheboundary,rela-
FM1
tivelytothetopologyofR1,ofπ c (C′ V1).
FM1 ∩
ThenC isCR-, -andLp-removable.
W
Thecondition c C ,whichisofcourseindependentofthechoiceofthetransversal
FM1{ }
R1 andofthethinneighborhoodV1,isillustratedinFIGURE 8of 5.1below;clearly,in
§
the case n = 2, it means that C V lies completely in one side of γ[ 1,1], with
′ 1
∩ −
respect to the topology of M1, as written in the statement of Theorem 1.2. Applying
someofourpreviousresultsinthisdirection([MP1],[MP3]),weshallprovideintheend
of Section 13 below some formulationsof applicationsof Theorem1.2’, close to being
analogsofTheorem1.3inhighercodimension.
Importantly,in order to let the geometriccondition c C appearless mysterious
FM1{ }
andtoarguethatitprovidestheadequategeneralizationofTheorem1.2tohighercodi-
mension, in the last Section 14 below, we shall describe an example of M, M1 and C
in C3 violating the condition c C , such that C is transversal to the characteristic
FM1{ }
foliationandistrulynonremovable.Thisexamplewillbeanalogousinsomesensetothe
exampleofanonremovableannulusdiscussedafterthestatementofTheorem1.2. Since
thereisnoH.Poincare´andI.Bendixsontheoremforfoliationsof3-dimensionalballsby
curves,itwillbeevenpossibletoinsurethatM andM1arediffeomorphictorealballsof
dimension4and3respectively. We maythereforeconcludethatTheorem1.2’provides
thedesirableanswertothe(alreadycitedsupra)Problem2.1raisedbyB.Jo¨rickein[J5],
p.432.
Topursuethepresentationofourresults,letuscommenttheassumptionthatM beof
codimension(n 1). Geometricallyspeaking,thestudyofclosedsingularitiesC lying
in a one-codime−nsional generic submanifold M1 of a generic submanifold M Cn
⊂
which is of CR dimension m 2 is more simple. Indeed, thanks to the fact that M1
≥
is of CR dimension m 1 1, there exist local Bishop discs completely attached to
− ≥
M1,andthishelpsmuchindescribingtheenvelopeofholomorphyofawedgeattached
to M C. On the contrary, in the case where M is of CR dimension 1, small analytic
\
8 JOE¨LMERKERANDEGMONTPORTEN
discsattachedtoamaximallyrealM1 are(trivially)inexistent. Thisiswhy,intheproof
of ourmain Theorems1.2 and 1.2’, we shall dealonly with small analytic discs whose
boundary is in part (only) contained in M1. Such discs are known to exist; we would
liketomentionthathistoricallyspeaking,thefirstconstructionofdiscspartiallyattached
to maximallyrealsubmanifoldswas exhibitedbyS. Pinchukin [P], whodevelopedthe
ideasofE.Bishop[B].
FinallywewilltestourmainTheorem1.2’inapplications. Firstofall,weclarifyits
relationtoknownremovabilityresultsinCRdimensiongreaterthanone. Herethemoti-
vationissimplythatmostquestionsofCRgeometryshouldbereducibletoCRdimension
1byslicing.Itturnsoutthatthemainknowntheoremsaboutremovablesingularities,due
toE.Chirka,E.L.Stout, andB.Jo¨rickeforhypersurfaces,andbytheauthorsinhigher
codimension([P1], Theorem1aboutLp-removability;[M2], Theorem3aboutCR-and
-removability)areallaratherdirectconsequenceofTheorem1.2’.Sincetheseresults
W
havenotyetbeenpublishedincompleteform,wetaketheoccasionofincludingthemin
thepresentpaper,asacorollaryofTheorem1.2’,yetdevisinganewgeometricapproach.
Theorem1.4. LetM bea 2,α-smoothgloballyminimalgenericsubmanifoldofCn of
C
CR dimension m 2 and of codimension d = n m 1, let M1 M be a 2,α-
smooth one-codim≥ensionalsubmanifold which is ge−neric≥in Cn and le⊂t C M1Cbe a
⊂
properclosedsubsetofM. Assumethatthefollowingconditionholds:
CR C : TheclosedsubsetC doesnotcontainanyCRorbitofM1.
OM1{ }
ThenC isCR-, -andLp-removable.
W
Noticethedifferencewiththecasem=1,wheretheanalogofCRorbitswouldconsist
ofcharacteristiccurves: thecondition c C doesnot saythatC shouldnotcontain
FM1{ }
any maximalcharacteristic curve. In fact, we observethat there cannotexist a uniform
removabilitystatementcoveringboththecasem=1andthecasem 2,whenceTheo-
≥
rem1.2’isstrongerthanTheorem1.4.Indeed,theelementaryexampleofanonremovable
circle inanannuluscontainedin theboundaryofastrictly convexdomainofC2 shows
thatC maybetrulynonremovablewhereasit doesnotcontainanycharacteristiccurve.
In the strictly pseudoconvexhypersurfacesetting, it is well known that Hopf’s Lemma
impliesthatboundariesofRiemannsurfacescontainedinC (andalsothetrackonC of
itsessentialhull,cf.[D])shouldbeeverywheretransversaltothecharacteristicfoliation.
Ofcourse,thisimpliesconverselythatC cannotcontainsuchboundaries(unlesstheyare
empty) if c C is satisfied. The reason why c C implies that C is removable
FM1{ } FM1{ }
alsointhenonpseudoconvexsettingandinarbitrarycodimensionwillbeappearantlater.
Finally,wementionthattheLp-removabilityofC inTheorem1.4holdsmoregenerally
withnoassumptionofglobalminimalityonM,asalreadynoticedin[J5],[P1],[MP1].
However,sincethecasewhereM isnotgloballyminimalessentiallyreducestothecon-
siderationofitsCR orbits,whicharegloballyminimalbydefinition,weshallonlydeal
withgloballyminimalgenericsubmanifoldsM throughoutthispaper.
Asafinalcomment,wepointoutthat,becausethepreviouslyknownproofsofTheo-
rem1.4wereoflocaltype,itissatisfactorytobringinthispaperapurelylocalframework
forthetreatmentofTheorems1.1,1.2,1.3and1.2’.
Our second group of applications concerns the classical edge of the wedge theo-
rem. Typicallyoneconsidersa maximallyrealedgeE to whichan opendoublewedge
( , ) is attached from opposite directions. One may interprete this configuration
1 2
W W
asapartialthickeningofagenericCRmanifoldM E containingE asa
1 2
⊂ ∪W ∪W
generichypersurface.Theclassicaledgeofthewedgetheoremstatesthatfunctionswhich
CHARACTERISTICFOLIATIONSANDENVELOPESOFHOLOMORPHY 9
arecontinuouson Eandholomorphicin extendholomorphicallyto
1 2 1 2
W ∪W ∪ W ∪W
aneighborhoodofE. Theorem1.2’impliesthatitsufficestoassumecontinuityoutsidea
removablesingularitiesofE. Thisallowsustoderiveanedgeofthewedgetheoremfor
meromorphicextension(Section13below).
This paper is divided in two parts: Part I contains the strategy per absurdumfor the
proofofTheorem1.2’,theconstructionofwhatwecallasemi-localhalf-wedgeandthe
choiceofaspecialpointtoberemovedlocally. PartIIcontainstheexplicitconstruction
offamiliesofhalf-attachedanalyticdiscs,theendofproofofTheorem1.2’andtheproofs
ofthevariousapplications.Thereaderwillfindamoredetaileddescriptionofthecontent
ofthepaperin 2.16below.
§
1.5. Acknowledgements. The authors would like to thank B. Jo¨ricke for several valu-
ablescientificexchanges. TheyacknowledgegeneroussupportfromtheEuropeanTMR
research network ERBFMRXCT 98063 and they also thank the universities of Berlin
(Humboldt),ofGo¨teborg(Chalmers),ofMarseille(Provence)andofUppsalaforinvita-
tionswhichprovidedopportunitiesforfruitfulmathematicalresearch.
2.DESCRIPTION OF THEPROOF OF THEOREM1.2AND ORGANIZATION OFTHE
§
PAPER
ThemainpartofthispaperisdevotedtotheproofofTheorem1.2’,whichwilloccupy
Sections 3, 4, 5, 6, 7, 8 and 9 below. In this preliminary section, we shall summarize
the hypersurfaceversionof Theorem1.2’, namelyTheorem1.2. Our goalis to provide
aconceptionaldescriptionofthebasicgeometricconstructions,whichshouldbehelpful
to read the whole paper. Because precise, complete and rigorous formulations will be
developedinthenextsections,weallowheretheuseofaslightlyinformallanguage.
2.1. Strategyper absurdum. Let M, S, and C be as in Theorem1.2. Itis essentially
knownthatboththeCR-andtheLp-removabilityofCarea(relativelymild)consequence
ofthe -removabilityofC (see 3.14andSection11below).Thus,weshalldescribein
W §
thissectiononlythe -removabilityofC.
W
First of all, as M is globallyminimal, it may be provedthat for everyclosed subset
C C, the complementM C is also globallyminimal(see Lemma 3.5 below). As
′ ′
M i⊂s of codimensionone in C\2, a wedge attached to M C is simply a connectedone-
sided neighborhood of M C in C2. Let us denote such\a one-sided neighborhoodby
\
ω . The goal is to prove that there exists a one-sided neighborhood ω attached to M
1
to whichholomorphicfunctionsinω extendholomorphically. By the definitionof -
1
W
removability,thiswillshowthatC is -removable.
W
Reasoningbycontradiction,weshalldenotebyC thesmallestnonremovablesubpart
nr
of C. By this we mean that holomorphic functions in ω extend holomorphicallyto a
1
one-sided neighborhood ω of M C in C2 and that C is the smallest subset of C
2 nr nr
\
such thatthisextensionpropertyholds. IfC is empty,theconclusionofTheorem1.2
nr
holds,gratuitously: nothinghastobeproved. IfC isnonempty,tocometoanabsurd,
nr
itsufficesto showthatatleastonepointofC islocallyremovable. By this, wemean
nr
thatthereexistsalocalone-sidedneighborhoodω ofatleastonepointofC suchthat
3 nr
holomorphicfunctionsinω extendholomorphicallytoω . Infact,thechoiceofsucha
2 3
pointwillbethemostdelicateandthemosttrickypartoftheproof.
In order to be in position to apply the continuity principle, we now deform slightly
M insidetheone-sidedneighborhoodω ,keepingC fixed,gettingahypersurfaceMd
2 nr
(with d like “deformed”) satisfying Md C ω . We notice that a local one-sided
nr 2
\ ⊂
10 JOE¨LMERKERANDEGMONTPORTEN
neighborhoodof Md at onepointp of C always containsa localone-sidedneighbor-
nr
hoodofM atp(thereadermaydrawafigure),sowe maywellworkonMd insteadof
working on M (however, the analogous property about wedges over deformed generic
submanifoldsisuntrueincodimension 2,see 3.16below,wheresupplementaryargu-
≥ §
mentsareneeded).
ReplacingthenotationC bythenotationC,thenotationMdbythenotationM and
nr
the notationω by the notation Ω, we see that Theorem1.2is reducedto the following
2
main proposition, whose formulation is essentially analogous to that of Theorem 1.2,
exceptthatitsufficestoremoveatleastonespecialpoint.
Proposition 2.2. Let M be a 2,α-smooth globally minimal hypersurface in C2, let
C
S Mbea 2,α-smoothsurfacewhichistotallyrealateverypoint.LetCbeanonempty
⊂ C
proper closed subset of S and assume that the nontransversalitycondition c C for-
mulatedinTheorem1.2holds.LetΩbeanarbitraryneighborhoodofM CFinSC{n.}Then
\
thereexistsaspecialpointp C andthereexistsalocalone-sidedneighborhoodω
sp ∈ psp
ofM inC2atp suchthatholomorphicfunctionsinΩextendholomorphicallytoω .
sp psp
2.3.Holomorphicextensiontoahalf-one-sidedneighborhoodofM. Thechoiceofthe
specialpointp willbeachievedintwomainsteps. Accordingtothenontransversality
sp
assumption c C ,thereexistsacharacteristicsegmentγ : [ 1,1] S withγ( 1)
FS{ } − → − 6∈
C, with γ(0) C and with γ(1) C such thatC lies in one (closed, semi-local) side
∈ 6∈
of γ in S. As γ is a Jordanarc, we mayorientS in M along γ, hencewe may choose
asemi-localopenside(S )+ ofS inM alongγ. Inthefirstmainstep(tobeconducted
γ
in Section 4 in the context of the general codimensional case Theorem 1.2’), we shall
construct what we call a semi-localhalf-wedge + attachedto(S )+ alongγ. By
HWγ γ
this,wemeanthe“halfpart”ofawedgeattachedtoaneighborhoodofthecharacteristic
segmentγinM,whichyieldsawedgeattachedtothesemi-localone-sidedneighborhood
(Sγ)+.Foranillustration,see FIGURE6below,inwhichoneshouldreplacethenotation
M1 by the notation S. Such a half-wedge may be interpreted as a wedge attached to
a neighborhoodof γ in S which is not arbitrary, but should satisfy a further property:
locally in a neighborhoodof every point of γ, either the half-wedge contains (S )+ or
γ
one of its two ribs contains(Sγ)+, as illustrated in FIGURE 6 below. Importantlyalso,
the conesofthisattachedhalf-wedgeshouldvarycontinuouslyaswe movealongγ, cf.
againFIGURE 6.
Thewayhowwewillconstructthishalf-wedge + isasfollows. Asillustratedin
HWγ
FIGURE 1justbelow,weshallfirstconstructastringofanalyticdiscsZr:s(ζ),whereris
theapproximateradiusofZ (∂∆),whoseboundariesarecontainedin(S )+ M and
r:s γ
⊂
whichtouchthecurveγ onlyatthepointγ(s),foreverys [ 1,1],namelyZ (1) =
r:s
∈ −
γ(s)andZ (∂∆ 1 ) (S )+.
r:s γ
\{ } ⊂