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 ©SpringerNatureSwitzerlandAG2019 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. 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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.