Inverse Spectral Geometry

InverseSpectralGeometryRobertBrooks12DepartmentofMathematicsHebrewUniversityGivatRam,Jerusalem,IsraelandDepartmentofMathematics,UniversityofSouthernCaliforniaLosAngeles,California90089-1113March,19951PartiallysupportedbyNSFgrant9200313andaFulbrightfellowship2Currentaddress:DepartmentofMathematics,TheTechnion,Haifa,IsraelInthispaper,wewouldliketosketchapictureaimedatgivingacom-prehensiveanswertothequestion:howdoesonegoaboutreconstructingamanifoldMfromthespectrumofitsLaplaceoperator?Itisunderstoodthat,ingeneral,thereisnouniquewayofreconstruct-ingM,becauseamanifoldisnotingeneraluniquelydeterminedfromitsspectrum.Soletusmakethefollowingdenition:Denition0.1AmanifoldMiscompactlydeterminedbyasetofconditionsP(whichMsatises)ifthereisanitesetfM1;:::;MkgandasetofmetricsM1;:::;MkonM1;:::;Mk,whicharecompactintheC1topology,suchthatanymanifoldM0whichsatisesPisisometrictoamanifoldlyinginoneoftheMi’s.Wethenhavethefollowing:Conjecture0.1Everycompactmanifoldiscompactlydeterminedbyitsspectrum.Wearestillfairlyfarawayfromthisconjectureinitsfullgenerality,althoughweremarkthatwecanobtaintheconjectureifweaddtothespec-trumsomecurvatureassumptionswhichinotherareasofgeometrywouldberegardedasfairlyweak.Ourfocusinthispaperwillbeonapresentationoftechniqueswhich,whenusedincombination,allowonetoattackthemainconjecture.Ourfeelingisthatthemaintechnicalcomponentsnecessarytoestablishthecon-jectureareinfairlygoodshape,andwewouldbesurprisedifaradicallydierentapproachwouldberequired,orevenhelpful,toarriveatthenaldestination.Withthatsaid,however,ineachsectiontherearetopicsandproblemswhichremainunexplored,andwhosesolutionwouldbeamajorsteptowardsthesolutionofthemainconjecture.Itisourpleasuretosetouthereourviewofwhattheseproblemsare.Ouremphasisherewillbeonsettingouthowvarioustechniquesareused,ratherthanhowtheyareproved,althoughwehavenotshiedawayfromsketchingaproofwhenwethoughtitwouldbeilluminating.Theplanofthepaperisasfollows:inx1,wegiveanoverviewofideasrelatedtotheCheegerFinitenessTheorem[Ch2]anditsgeometricrelatives.Thisisthemaintechniquebywhichonebuildsupa\roughmodelofaman-ifoldfromgeometricdata.Inx2,wediscussbootstrappingtechniques.These1techniquesservetwoimportantpurposes:rstly,theyallowoneto\smoothouttheroughmodelsofx1.Secondly,byexaminingwhatnecessaryinputisrequiredtokeepthebootstrapmachineryrunning,wegetagoodpictureofwhatkindsofgeometricdataweneedtoextractfromthespectrum.Finally,inx3weaddresstheproblemofactuallyextractingthedesiredgeometricdatafromthespectrum.Acknowledgements:ItisapleasuretothankStigAnderssonforhisgeneroushospitalityduringthecourseoftheSummerSchoolinInverseSpec-tralGeometry,andinparticularforhislast-minuteeortswhichallowedmetoparticipateonshortnotice.Iwouldalsoliketothankhimforhisveryilluminatingsuggestionsonthepossiblescopeanddirectionofthepresentpaper.Finally,Iwouldliketotakethisopportunitytothankmycolleagues,includingthosepresentattheconferenceandthosewhocouldnotattend,formakingspectralgeometryatrulypleasantandexcitingareainwhichtowork.Whileititmyhopethatthepicturepresentedherewillinducesometojointhisareaofresearch,Ithinkthatafargreaterinducementwouldbetheopportunitytogettoknow,andtobeapartof,thecommunitywhichoccupiesitselfwiththesequestions.1CheegerFinitenessTherstquestiononemustdealwithinattemptingtoreconstructamani-foldfromitsspectrumis:whatkindofpropertiesarerequiredtocompactlydetermineamanifold?Ineect,theinversespectralproblemismadecon-siderablyeasieronceonehassomekindofmodelspaceonwhichtowork,althoughevenheretherearemanyinterestingandchallengingproblems.Sowewillaskthequestion:whatkindofmaterialisrequiredtobuildsuchamodelspace?Thesolutiontothisproblemindimension2isquitestandard,andgoesbacktoMcKeanandSinger[MS]:thea1termintheheatexpansionis,uptoanon-zeroconstant,theintegraloverMofthescalarcurvature.BytheclassicalGauss-BonnetTheorem,thisdeterminestheEulercharacteristicofM,andhence,intheorientablecase,thedieomorphismtypeofM.Fordimensionsbiggerthan2,thetopologicalsituationisfarmorecom-plicated.Forinstance,eveninthecaseofmanifoldswithconstantcurvature21,thereexistmanifoldsMiwhosevolumesaccumulateatsomenitevalue(thevolumeofahyperbolicmanifoldwithcusps),suchthattheMi’shavedierentfundamentalgroups.Itfollowsthata1(orevena2;a3;:::)willnotbeabletodierentiatebetweenslightlyperturbedversionsoftheMi’s.WeliketothinkofCheeger’sFinitenessTheorem[Ch2]ashavingtwoparts:aphilosophicalpartandatechnicalpart.Thephilosophicalpartisastatementthatsomereasonablecollectionofgeometricpropertieswilldetermineaclassofmanifoldsuptonitelymanypossibilities.Thetechnicalpartthenaskstowhatextentonecanreducethelistofgeometricpropertiestoaminimum.HereisastatementofaphilosophicalversionoftheCheegerFinitenessTheorem:Theorem1.1([Ch2])Forpositivenumbersv;V;inj;andD,andarbitraryrealnumberskandK,letfMgbeacollectionofmanifoldssatisfying:(i)vvol(M)VforMfMg.(ii)kK(M)K;whereK(M)runsoverallsectionalcurvaturesinM,forMfMg.(iii)inj(M)inj,forMfMg.ThenfMgcontainsonlynitelymanydieomorphismtypes.ItwasobservedbyGromov([Gr],seealso[Kas],[GW],[P]forproofs