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Tutorials, Schools, and Workshops in the Mathematical Sciences Vicente Cortés Klaus Kröncke Jan Louis Editors Geometric Flows and the Geometry of Space-time Tutorials, Schools, and Workshops in the Mathematical Sciences The seriesis the primaryresourcefor the publicationof resultsand developments presentedatsummerorwinterschools,workshops,tutorials,andseminars.Written in an informal and accessible style, they present important and emerging topics in scientific research for PhD students and researchers. Filling a gap between traditional lecture notes, proceedings, and standard textbooks, the titles included inTSWMSpresentmaterialfromtheforefrontofresearch. Moreinformationaboutthisseriesathttp://www.springer.com/series/15641 ¨ Vicente Cortés • Klaus Kroncke • Jan Louis Editors Geometric Flows and the Geometry of Space-time Editors VicenteCortés KlausKro¨ncke DepartmentofMathematics DepartmentofMathematics UniversityofHamburg UniversityofHamburg Hamburg,Germany Hamburg,Germany JanLouis DepartmentofPhysics UniversityofHamburg Hamburg,Germany ISSN2522-0969 ISSN2522-0977 (electronic) Tutorials,Schools,andWorkshopsintheMathematicalSciences ISBN978-3-030-01125-3 ISBN978-3-030-01126-0 (eBook) https://doi.org/10.1007/978-3-030-01126-0 LibraryofCongressControlNumber:2018960861 MathematicsSubjectClassification(2010):53C44,53C50,53C27,53C29 ©SpringerNatureSwitzerlandAG2018 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. This book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered companySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Preface Amajorgoalinmathematicsaswellasinphysicshasbeenandstillistounderstand the geometry of space and time. Developments in both subjects have fruitfully influenced each other over the history of science. The formulation of general relativitybyEinsteinwouldnothavebeenpossiblewithouttheconceptsof(semi-) RiemanniangeometrythathademergedwiththevisionaryideasofRiemanninthe previouscentury.Conversely,ideasfromgeneralrelativityinfluencedmathematical research and the study of Einstein’s equation is one of today’s major topics in geometricanalysis. Similarly, the development of more recent areas of theoretical physics, such as string theory, is deeply connected to the study of geometric problems in mathematics, such as the study of metrics of special holonomy.It turned out that geometricflowsarealsoofgreatimportanceintheinterplaybetweenmathematics andphysics;e.g.,theRiemannPenroseinequalityhasbeenshownbyHuiskenand Ilmanenusingtheinversemeancurvatureflow. This volume is based on a summer school and workshop entitled “Geometric flows and the geometry of space-time” held at the University of Hamburg in September 2016. The aim of this event was to provide a forum where physicists and mathematicians can exchange ideas and where graduate students and young researchersgettheopportunitytolearnaboutrecentdevelopmentsattheintersection ofmathematicsandphysics. It brought together around 60 participants with mathematical and physical backgrounds.ThespeakerswereLarsAndersson,HelgaBaum,SpirosCotsakis,Pau Figueras,GaryGibbons,MarkHaskins,JasonLotay,ThomasLeistner,JanMetzger, andOliverC.Schnürer. Out of these 10 speakers, 7 gave two talks where the first one was more of an introductorynatureandtheotheronewasmorefocusedonactualresearch.These talks covereda broadvariety of topics, rangingfromspecial holonomymetricsto variousconceptsofmassingeneralrelativityandthenumericalandanalyticstudy ofblackholespace-times. Moreover,threeofthespeakersgaveminicourseswhereeachofthemhadatotal length of 180min. One minicourse was more of a physical nature and was held v vi Preface by Gary Gibbons about the theory of black holes. The other two lecture courses were more of a mathematical nature. One course was held by Oliver C. Schnürer aboutgeometricflowsandfocusedinparticularonmeancurvatureflow.Theother course held by Helga Baum was about special holonomy and parallel spinors in Lorentziangeometry.Inaddition,wehadtworelatedtalksaboutCauchyproblems forLorentzianmanifoldsofspecialholonomybyThomasLeistner. This volume consists of two articles. The first is based on the mathematical lecture course by Oliver C. Schnürer and the second on the mathematical lecture course by Helga Baum extended by results presented in the lectures by Thomas Leistner. Anothervolumebasedonthethirdlecturecourseaboutthetheoryofblackholes is planned. The papers are written for graduate students and researchers with a generalbackgroundingeometryandinthetheoryofpartialdifferentialequations, who want to get acquainted with these central subjects of modern geometry. We hopethisvolumewillbehelpfulandinspiring. Hamburg,Germany VicenteCortés July2018 KlausKröncke JanLouis Contents LorentzianGeometry:Holonomy,Spinors,andCauchyProblems ........ 1 HelgaBaumandThomasLeistner GeometricFlowEquations...................................................... 77 OliverC.Schnürer vii Lorentzian Geometry: Holonomy, Spinors, and Cauchy Problems HelgaBaumandThomasLeistner Contents 1 Introduction..................................................................................... 2 2 BasicNotions................................................................................... 4 3 LorentzianHolonomyGroups................................................................. 7 3.1 BasicsonHolonomyGroups............................................................ 7 3.2 HolonomyGroupsofLorentzianManifolds............................................ 14 4 LorentzianSpinGeometry:CurvatureandHolonomy....................................... 20 4.1 SpinStructuresandSpinorFields....................................................... 21 4.2 CurvatureandHolonomyofLorentzianManifoldswithParallelSpinors............ 25 5 ConstraintEquationsforSpecialLorentzianHolonomy..................................... 34 5.1 ConstraintEquationsforRecurrentandParallelVectorFields........................ 34 5.2 ConstraintEquationsforParallelSpinorFields........................................ 37 6 TheCauchyProblemfortheVacuumEinsteinEquations ................................... 42 6.1 TheConstraintConditionsfortheVacuumEinsteinEquations ....................... 43 6.2 ResultsfromPDETheory............................................................... 44 6.3 TheVacuumEinsteinEquationsasEvolutionEquations.............................. 46 6.4 TheVacuumEinsteinEquationsasSymmetricHyperbolicSystem................... 49 7 CauchyProblemsforLorentzianSpecialHolonomy......................................... 55 7.1 EvolutionEquationsforaParallelLightlikeVectorFieldintheAnalyticSetting.... 56 7.2 TheCauchyProblemforaParallelLightlikeVectorFieldasaSymmetric HyperbolicSystem....................................................................... 59 7.3 CauchyProblemforParallelLightlikeSpinors......................................... 65 8 GeometricApplications........................................................................ 66 8.1 ApplicationstoLorentzianHolonomy.................................................. 67 8.2 ApplicationstoSpinorFieldEquations................................................. 72 References.......................................................................................... 73 H.Baum((cid:2)) HumboldtUniversityBerlin,DepartmentofMathematics,Berlin,Germany e-mail:[email protected] T.Leistner((cid:2)) SchoolofMathematicalSciences,UniversityofAdelaide,Adelaide,SA,Australia e-mail:[email protected] ©SpringerNatureSwitzerlandAG2018 1 V.Cortésetal.(eds.),GeometricFlowsandtheGeometryofSpace-time,Tutorials, Schools,andWorkshopsintheMathematicalSciences, https://doi.org/10.1007/978-3-030-01126-0_1 2 H.BaumandT.Leistner Abstract ThisreviewisbasedonlecturesgivenbytheauthorsduringtheSummer School Geometric Flows and the Geometry of Space-Time at the University of Hamburg, September 19–23, 2016. In the first part we describe the algebraic classification of connectedLorentzianholonomygroups.In particular,we specify theholonomygroupsoflocallyindecomposableLorentzianspinmanifoldswith a parallelspinorfield.Inthesecondpartweexplainnewmethodsfortheconstruction of globally hyperbolic Lorentzian manifolds with special holonomy based on the solutionofcertainCauchyproblemsforPDEsthatareimposedbytheexistenceofa parallellightlikevectorfieldoraparallellightlikespinorfieldwithinitialconditions onaspacelikehypersurface.Thereby,wederiveasecondorderevolutionequation ofCauchy-Kowalevskitypethatcanbesolvedintheanalyticsettingaswellasan appropriate first order quasilinear hyperbolic system that yields a solution in the smoothcase. 1 Introduction This review is based on lectures given by the authors during the Summer School Geometric Flows and the Geometry of Space-Timeat the University of Hamburg, September 19–23, 2016. In these lectures we described at one hand the algebraic classificationofconnectedLorentzianholonomygroupsandexplainedattheother hand new methods for the construction of Lorentzian manifolds with special holonomy based on the solution of appropriate Cauchy problems with initial conditionsonaspacelikehypersurface. The holonomy group of a semi-Riemannian manifold (M,g) is the group of paralleltransportsalongallcurvesthatareclosedatafixedpointx ∈M.ItisaLie subgroupofthegroupofallorthogonaltransformationsof(T M,g ),itsconnected x x componentisisomorphictotheholonomygroupoftheuniversalsemi-Riemannian coveringof(M,g). Theconceptofholonomywasprobablyfirstsuccessfullyappliedindifferential geometry by E. Cartan [31–33], who used it to classify symmetric spaces. Since then,ithasprovedtobeaveryimportantconcept.Inparticular,itallowstodescribe parallelsectionsingeometricvectorbundlesover(M,g)—suchastangent,tensor orspinorbundles—asholonomyinvariantobjectsandthereforebypurelyalgebraic tools.Moreover,geometricpropertieslikecurvaturepropertiescanbereadoffifthe holonomygroupisspecial,i.e.,apropersubgroupofO(T M,g ).Oneoftheimpor- x x tant consequencesof the holonomynotion is its application to the ‘classification’ of special geometriesthat are compatible with Riemannian geometry.For each of thesegeometriesanownbranchofdifferentialgeometryhasdeveloped,forexample Kähler geometry (holonomy U(m)), geometry of Calabi-Yau manifolds (SU(m)), hyper-Kählergeometry(Sp(k)), quaternionicKähler geometry (Sp(k)·Sp(1)), or theexceptionalgeometryofG -manifoldsorofSpin(7)-manifolds.Inphysicsthere 2 is muchinterestin semi-Riemannianmanifoldswith specialholonomy,since they