Holomorphic Curves and Global Questions in Contact Geometry

328 Pages·2019·3.555 MB·English

by  Casim Abbas, Helmut Hofer

Birkhäuser Advanced Texts
Basler Lehrbücher
Casim Abbas
Helmut Hofer
Holomorphic
Curves and Global
Questions in
Contact Geometry

BirkhäuserAdvancedTextsBaslerLehrbücher
Serieseditors
StevenG.Krantz,WashingtonUniversity,St.Louis,USA
ShrawanKumar,UniversityofNorthCarolinaatChapelHill,ChapelHill,USA
JanNekováˇr,UniversitéPierreetMarieCurie,Paris,France
Moreinformationaboutthisseriesathttp://www.springer.com/series/4842
Casim Abbas (cid:129) Helmut Hofer
Holomorphic Curves
and Global Questions
in Contact Geometry
CasimAbbas HelmutHofer
MichiganStateUniversity InstituteforAdvancedStudy
EastLansing,MI,USA Princeton,NJ,USA
ISSN1019-6242 ISSN2296-4894 (electronic)
BirkhäuserAdvancedTextsBaslerLehrbücher
ISBN978-3-030-11802-0 ISBN978-3-030-11803-7 (eBook)
https://doi.org/10.1007/978-3-030-11803-7
LibraryofCongressControlNumber:2019930036
MathematicsSubjectClassification(2010):58-xx,37-xx,32-xx
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DedicatedtotheMemoryofAndreas Floer
andKris Wysocki
AndreasFloer
Author:GeorgeM.Bergman.
Source:ArchivesoftheMathematisches
ForschungsinstitutOberwolfach
KrisWysocki
Author:JürgenPöschel.
Source:ArchivesoftheMathematisches
ForschungsinstitutOberwolfach
Introduction
Historical Background
In1976,inhispaper[91],JürgenMoserwritesabouttheclassicalactionprinciple
thefollowing:“However,thisvariationalprincipleisverydegenerate,forexample
eventheLegendreconditionisviolated,andiscertainlynotsuitableforanexistence
proof.” If this statement would be true, modern symplectic geometry would not
exist.Inessence,Moserclaimed(forwhathethoughtwereobviousreasons)thata
certainformalvariationalprinciplecannotbeusedtofindglobalperiodicorbitsfor
finite-dimensional Hamiltonian systems. In today’s language, it was an argument
againsttheexistenceofFloertheory.Moser’sstatementwaswell-foundedinwhat
was known at the time. Morse theory in finite dimensions was well-understood,
andstarting with the workofPalais andSmale, the ideaswere appliedto infinite-
dimensionalproblems.However,abasicassumption,besideswhatiscallednowthe
Palais-Smalecondition,wasthatMorseindicesofaMorsefunctionf werefinite,
whereastheco-indexwasallowedtobeinfinite(inthereversedcaseonecanstudy
−f).FromMorsetheory,oneknewthatpassinganon-degeneratecriticalpoint,the
homotopytypeofthesub-levelsetschangedbyaddingafinite-dimensionalhandle
of the appropriate dimension. In the case that the Morse index was infinite, the
homotopytypewouldnotchangeduetothefactthattheunitsphereinaninfinite-
dimensional Hilbert space is contractible. As a consequence, the global existence
mechanism for finding critical points by studying the change of homotopy type
wouldnotwork.This, in a nutshell, was Moser’sargument:if the topologyof the
sub-levelsetswouldneverchange,howcouldtherebeaglobalmechanismtofind
them?
The argument is undoubtedly correct for general functionals, but it does not
apply to the classical action functional.Indeed,shortly afterward,Moser’s former
student P. Rabinowitz showed in his paper [95] that a nonlinear wave equation
problem, an infinite-dimensional Hamiltonian system, has in fact periodic orbits,
byusingageneralizationoftheclassicalactionprinciple.AsJ.Moseroncetoldthe
second author (HH), this paper clearly provedthat he was mistaken and he asked
vii
viii Introduction
P.Rabinowitzifhecouldusehismethodstoprovetheexistenceofperiodicorbits
infinite-dimensionalHamiltoniansystems.ThisresultedinRabinowitz’celebrated
papers[96]and[97]wherethelatterpromptedtheinfluentialWeinsteinconjecture
[114]. A first breakthrough on the Weinstein conjecture was Viterbo’s result that
everyregularcompactenergysurfaceinR2ncarriesaperiodicorbit(see[112]).This
ledHHandZehndertoexhibitanimportantphenomenoninHamiltoniandynamics
called almost existence (see [65], and for a sharpening of the result, Struwe’s
contribution[107],andmoregenerally[66]fora“symplecticcapacityviewpoint”).
ThenotionofsymplecticcapacitywasintroducedbyEkelandandHHintheirpapers
[27, 28]. The basic observation was that Gromov’s non-squeezing result, which
was proved by pseudoholomorphicmethods, is related to an infinite class of new
monotonicsymplecticobstructionswhicharerelatedtoHamiltoniandynamicsand
associatedspectralproperties.Thesepaperspavethewayforsymplectichomology
[20,21,48,49].
Rabinowitz’s papers are the starting point for considerable research activities.
Particularlyimportantare[9]which,togetherwithConley’sindextheory[23],lays
thefoundationforthebreakthroughresult[24],establishingtheArnoldconjecture
forfixedpointsofHamiltoniandiffeomorphismsonstandardsymplectictoriandthe
significantcontributionbyEkelandandLasryonthenumberofperiodicorbitsona
convexenergysurface[29](seealso[26]forsomeofthesubsequentdevelopments).
A preprint by Eliashberg also addressing the abovementioned Arnold conjecture
[30]wasunknowntomostresearchersintheWest.
Another important development was Bennequin’s Ph.D. thesis [13], in which
amongotherthings,anexoticcontactstructureonR3isconstructed.Thisisthefirst
exampleofwhatislatercalledanovertwistedcontactstructure.Bennequin’swork
can also be seen as the starting point of Eliashberg’sstriking work on the subject
[31–33].
In1985,[60],Gromovintroducesthetheoryofpseudoholomorphiccurves.Soon
thereafter,thepseudoholomorphiccurveideaiscombinedwiththeConley-Zehnder
variational viewpoint by A. Floer in a series of seminal papers [43–47], which
subsequentlyopensupdifferentareasofmathematics.
Relevant for this book is the following development. In [64], HH develops a
version of Gromov’s pseudoholomorphiccurve theory [60], applicable to contact
manifoldsandasanapplicationthatsolvesmanycasesoftheWeinsteinconjecture.
Thepaper[64]makesaconsiderableuseoftwoofthebefore-mentionedpapersby
Eliashberg,namely,[31,32].
AboutThisBookandIts Context
In some sense, this book is a “classic.” It grew out of a NachdiplomVorlesung at
theETHZürichgivenbyHHduringtheacademicyear1993/1994.Thelecturewas
builtaroundthepaper[64]andwasalmostindefiniteformin1995.Thekeyresult
provedinthistextshowsthateveryReebvectorfieldonS3 hasatleastoneperiod
Introduction ix
orbit,establishingtheWeinsteinconjectureforthethree-sphere.Unfortunately,the
authorsneverfoundthe2monthstofinishtheprojectuntilrecently.However,quite
a number of students learned the material through the preliminary drafts floating
around.Thematerialprovidesaninterestingentrypointtothepseudoholomorphic
curvetheorybyprovingsomeofthedeeperresultsinthesubject.Ithasbeenused
by the authors for graduate courses on the subject, and the assumed prerequisites
aremodest,andthegivenproofsareverydetailed.
Thebookcanbeviewedasanentrypointforthestudyoffiniteenergyfoliations
(see[68,69]),whichinitiallywasusedtostudylow-dimensionaldynamicsbutlater
evolved as a tool for studying low-dimensional symplectic problems (see [115]).
Thebookisalsoanentrypointtosymplecticfieldtheory(SFT)[36].Resultsabout
the Weinstein conjecture for planar contact structures as proved in [3] arise from
thesetwoviewpoints.TheresultsbyHryniewiczandcollaborators(see[70–73])are
fundamentalcontributionstothetheoryofglobalsurfacesofsection,whichhavea
precisionwhich20yearsagowouldhavebeenunimaginable.Duetoarecentresult
ofJoelFishandHH[39],themethodsgainadditionalimportancesincetheycanbe
usedinasuitablemodificationtoprovetheexistenceofso-calledferalcurves.These
canbeviewedasastronggeneralizationofpseudoholomorphiccurvesandcanbe
usedtoderivedeepresultsinsymplecticdynamics(see[18]).Symplecticdynamics
is an emergingfield with an already impressive number of importantresults (see,
e.g.,[4,16,17,56,75]). The recentdevelopmentof feralcurvesallowsto answer
a 20-year-oldquestionby the dynamicistMichelHerman,raised at his 1998ICM
address,inthe specialcase ofdimensionfour[61].Moreprecisely,an application
ofthetheoryofferalcurvesconfirmsthata compactregularenergysurfaceinthe
standardsymplecticR4 cannotharboraminimalflow,i.e.,a flowforwhichevery
orbitisdense.Theresultstronglyhintstowarda theoryofalgebraicinvariantsfor
closed invariant subsets of Hamiltonian flows substituting for invariants build on
periodic orbits. Of course, this is a question for future research, and the outcome
willhaveramificationsforthedevelopmentofthefield.
Whatto Do Next?
Inthelast40years,symplecticgeometrydevelopedintoahugefield,anditissurely
challengingforthenovicetofindher/hisway.Assumingthereaderhasdigestedthe
currenttext,thereareseveraldirectionsforfurtherstudy.Sinceweonlyprovideda
minimum of background material, it is clearly advantageous to learn more about
contact and symplectic geometry. A very useful reference for contact geometry
is Geiges’ book [54]. However, there have been recent developments around the
so-called hard vs soft dichotomy, and we refer the reader to the survey article by
Eliashberg [35] and the references discussed in this survey. We also recommend
the importantbookby Cieliebak and Eliashberg[19] which is concernedwith the
symplectic geometry of affine complex manifolds and in which contact geometry
playsan importantrole.These referencesprovidethe geometricbackboneofa lot
x Introduction
of current research, providedone would like to venture into the geometric realm.
Manyofthegeometricproblemsrequiresomehardtechnology,i.e.,knowledgeof
pseudoholomorphiccurvetheory,particularlywhentheyusesymplecticorcontact
invariants. Another important source of background material are the books by
McDuffandSalamon[86,87].
Focusing back on pseudoholomorphic curve theory, there are many different
directions which lead to get a deeper understanding of this theory. Compactness
resultsareanimportantsteppingstone(see[2,15]and[22]).Theapproachesdiffer
somewhatandareinlinewiththeseminalworkofGromov[60]butstudythemore
complicatedcaseofsymplecticcobordismswithcontacttypeorstableHamiltonian
ends.JoelFish[40,41]takesaquitedifferentviewpointinhisfoundationalpapers
oncompactnesspropertiesofpseudoholomorphiccurves.Hisworkprovidesavery
powerful new take on the original compactness ideas, and the two papers are the
obvious starting point for further developments. In particular, this work prepares
verywellforastudyof[39,42].Thesetwopapersarethestartingpointforthenew
classof“feralcurves”andtheirapplicationsinsymplecticdynamics.
Anotherimportantbasic topicisSiefring’sintersectiontheory(see [102,103]).
After this, one can study finite energy foliations (see [68, 69]). On the more
geometricside,therearemanyapplicationsaimingatthestudyoflow-dimensional
symplectic and contact manifolds, and we refer the reader for the further devel-
opments to [115]. If one wants to focus on the dynamical applications, they are
plentifulas well. Particularly,the classical restrictedthree-bodyproblemis a very
interesting subject of study, and we refer the reader to [50] and its numerous
references.
Another interesting direction is concerned with embedded contact homology
(ECH),andtherearestillmanyopenproblems.Thatthistheoryisverypowerfulcan
beeasilyinferredfromtheresultswhichhavebeenobtained.ECHiscloselyrelated
to Seiberg-Witten-Floertheory,and manyof its propertiescan only be derivedby
this connection. Historically, motivated by the results of Taubes about Seiberg-
Witten theory and the event of SFT, M. Hutchings [74] predicted a relationship
between Seiberg-Witten-Floer theory and a version of SFT called embedded
contacthomology(ECH).Thistheoryhasstrikingapplicationsinlow-dimensional
symplectic geometryand symplectic dynamics,i.e., Irie used it in [75] to provea
smoothclosinglemma.
Acknowledgements HHwouldliketothankhis“fellowtravelers,”IvarEkeland,YashaEliash-
berg, Dusa McDuff, Dietmar Salamon, Claude Viterbo, and Edi Zehnder, for many interesting
discussions, comments, andsuggestions andmost ofallfortheirfriendship. Andreas Floer and
KrisWysocki,whowerepartofthisjourney,aregreatlymissed.