Trends in Mathematics Research Perspectives CRM Barcelona Vol.3 Maria del Mar González Paul C. Yang Nicola Gambino Joachim Kock Editors Extended Abstracts Fall 2013 Geometrical Analysis Type Theory, Homotopy Theory and Univalent Foundations Trends in Mathematics Research Perspectives CRM Barcelona SeriesEditors EnricVentura AntoniGuillamon Since1984theCentredeRecercaMatemàtica(CRM)hasbeenorganizingscientific eventssuchasconferencesorworkshopswhichspanawiderangeofcutting-edge topicsinmathematicsandpresentoutstandingnewresults. Inthefallof2012,the CRM decidedto publishextendedconferenceabstractsoriginatingfrom scientific eventshostedatthecenter.Theaimofthisinitiativeistoquicklycommunicatenew achievements,contributeto a fluentupdateofthe state of theart, andenhancethe scientificbenefitoftheCRMmeetings.Theextendedabstractsarepublishedinthe subseriesResearchPerspectivesCRMBarcelonawithintheTrendsinMathematics series.Volumesinthesubserieswillincludeacollectionofrevisedwrittenversions ofthecommunications,groupedbyevents. Moreinformationaboutthisseriesathttp://www.springer.com/series/4961 Extended Abstracts Fall 2013 Geometrical Analysis Maria del Mar González Paul C. Yang Editors Type Theory, Homotopy Theory and Univalent Foundations Nicola Gambino Joachim Kock Editors Editors MariadelMarGonzález PaulC.Yang DepartamentdeMatemaJticaAplicada DepartmentofMathematics UniversitatPoliteJcnicadeCatalunya PrincetonUniversity Barcelona,Spain Princeton,NJ,USA NicolaGambino JoachimKock SchoolofMathematics DepartamentdeMatemaJtiques UniversityofLeeds UniversitatAutoJnomadeBarcelona Leeds,UnitedKingdom Barcelona,Spain ISSN2297-0215 ISSN2297-024X (electronic) TrendsinMathematics ISBN978-3-319-21283-8 ISBN978-3-319-21284-5 (eBook) DOI10.1007/978-3-319-21284-5 LibraryofCongressControlNumber:2015955730 MathematicsSubjectClassification(2010):Firstpart:53Axx;Secondpart:55Pxx SpringerChamHeidelbergNewYorkDordrechtLondon ©SpringerInternationalPublishingSwitzerland2015 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. 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Printedonacid-freepaper Springer International Publishing AG Switzerland is part of Springer Science+Business Media (www.birkhauser-science.com) Contents PartI GeometricalAnalysis A Positive Mass Theorem in Three Dimensional Cauchy–RiemannGeometry.................................................... 3 Jih-HsinCheng,AndreaMalchiodi,andPaulYang OntheRigidityofGradientRicciSolitons.................................... 9 ManuelFernández-LópezandEduardoGarcía-Río Geometric StructuresModeled on Affine Hypersurfaces andGeneralizationsoftheEinstein–WeylandAffineSphere Equations ......................................................................... 15 DanielJ.F.Fox SubmanifoldConformalInvariantsandaBoundaryYamabeProblem... 21 A.RodGoverandAndrewWaldron VariationoftheTotalQ-PrimeCurvatureinCRGeometry................ 27 KengoHirachi Conformal Invariants from Nullspaces of Conformally InvariantOperators.............................................................. 31 DmitryJakobson RigidityofBach-FlatManifolds ............................................... 35 SeongtagKim UniformizingSurfaceswithConicalSingularities............................ 41 AndreaMalchiodi RecentResultsandOpenProblemsonConformalMetrics onRnwithConstantQ-Curvature ............................................. 45 LucaMartinazzi v vi Contents Isoperimetric Inequalities forComplete ProperMinimal SubmanifoldsinHyperbolicSpace............................................. 51 Sung-HongMinandKeomkyoSeo TotalCurvatureofCompleteSurfacesinHyperbolicSpace................ 57 JunO’HaraandGilSolanes ConstantScalarCurvatureMetricsonHirzebruchSurfaces............... 61 NobuhikoOtoba IsoperimetricInequalitiesforExtremalSobolevFunctions................. 67 JesseRatzkinandTomCarroll PartII Type Theory, HomotopyTheory, and Univalent Foundations UnivalentCategoriesandtheRezkCompletion.............................. 75 BenediktAhrensm,KrzysztofKapulkin,andMichaelShulman CoveringSpacesinHomotopyTypeTheory.................................. 77 Kuen-BangHou TowardsaTopologicalModelofHomotopyTypeTheory................... 83 PaigeNorth Made-to-OrderWeakFactorizationSystems................................. 87 EmilyRiehl ADescentPropertyfortheUnivalentFoundations .......................... 93 EgbertRijke ClassicalFieldTheoryviaCohesiveHomotopyTypes....................... 99 UrsSchreiber HowIntensionalIsHomotopyTypeTheory?................................. 105 ThomasStreicher Part I Geometrical Analysis Editors MariadelMarGonzález PaulC.Yang Foreword In this Part of the presentvolume of the Birkhauser series Research Perspectives CRMBarcelonawepresent13ExtendedAbstractscorrespondingtoselectedtalks given by participants in the Geometric Analysis conference that took place at the CentredeRecercaMatemàtica(CRM)fromJuly1stto5th,2013.Thisconference was a central part of the Intensive Research Programme on Conformal Geometry and Geometric PDE’s that took place at the CRM during the summer of 2013. Theresultspresentedinthisvolumeconstituteabriefoverviewofcurrentresearch in the field of Geometric Analysis. This modern field lies at the intersection of many branches of mathematics (Riemannian, Conformal, Complex or Algebraic Geometry,Calculus of Variations,PDE’s, etc.) and relates directly to the physical worldsincemanynaturalphenomenapossesanintrinsicgeometriccontent. Conformal geometry is the study of the set of angle-preserving (conformal) transformationsonaspace.Whileintwodimensionsthisispreciselythegeometry ofRiemannsurfaces,indimensionsthreeandabovethisstudyopensupmanynew differentsubjects,leadingtotheverywidefieldnamedconformalgeometry. The first question is to find conformal invariants or, more specifically, confor- mally covariantoperators,that is, operatorswhich satisfy some invariantproperty under conformal change of metrics on a manifold, and its associated curvature. The model example is the Laplace–Beltrami operator, in relation to the Yamabe problem.The Yamabeequationis a second order,semilinear PDE; we would like to understandhigherorder or fully non-lineargeneralizations,such as the Paneitz operators together with Q-curvature, or the (cid:2) equation. As a consequence, new k 2 I GeometricalAnalysis interestingdirectionsinPDE’shavebeenopenedup,whereexistenceorregularity theoryisnotdevelopedasmuch.Lately,therehasbeenalotofinterestinthestudy of non-local,conformallycovariantoperatorsof fractionalorder constructedfrom Poincaré–Einstein metrics. While they are natural objects in other areas such as probability,theirgeometricalmeaningisnotyetwellunderstood. Particularly,thestudyofPoincaré–Einsteinmetricshasbeenandcontinuestobe a rich source of activity relating conformal and Riemannian geometry. These are complete Einstein metrics which are asymptotically hyperbolic at infinity. Their boundary at infinity invariantly inherits a conformal structure. The asymptotic behavior of the metric encodes a great deal of information about the conformal structureatinfinity,andthishasledtonewconstructionsandprogressinconformal geometry. On the other hand, there are many analytic problems concerning the existence, uniqueness and regularity of Poincaré–Einstein metrics with a given conformalinfinityandplentyofopenquestions.Thistopicisstimulatedbyitsrole intheAdS/CFTcorrespondenceinPhysics. In CR geometry there are formal similarities with conformal geometry. For example, there are conformally covariant operators analogous to the conformal LaplacianandthePaneitzoperators.Whiletheseoperatorsalsocomewithassoci- atedQ-curvaturequantities,theirgeometric/analyticmeaningisquitedifferentfrom conformalgeometry.Theanalysisofthese operatorsis closelyconnectedwiththe geometryofthepseudoconvexmanifoldswhichtheymaybound,henceofinterest inseveralcomplexvariables. Finally, one of the classical topics in Geometric Analysis is the study of variationalproblemsrelated tothe functionalarea.In thissense, the globaltheory ofminimalandconstantmeancurvaturesurfacesinhomogeneousthree-manifolds, andmoregenerallyinRiemannianandsub-Riemannianmanifolds,representstoday atremendouslyactivefieldofnewdiscoveriesandchallenges.Thelocalmodelsin sub-RiemannianGeometryaretheCarnotgroups,withaspecialroleplayedbythe Heisenberggroup.Applicationsof minimalsurfacesto other subjects includelow dimensional topology, general relativity and materials science. Closely related to this topic appearsthe isoperimetricproblem,connectingGeometricAnalysis with GeometricMeasureTheory. The editors would like to thank the supportof the CRM in the organizationof thisresearchprogramme.Wehopeitservesasaninspirationforfuturedirectionsin thefield. Barcelona,Spain MariadelMarGonzález Princeton,NewJersey,USA PaulC.Yang A Positive Mass Theorem in Three Dimensional Cauchy–Riemann Geometry Jih-HsinCheng,AndreaMalchiodi,andPaulYang In this note we summarize the results from [6] on the positive mass problem in 3-dimensionalCR(Cauchy–Riemann)geometry. Weconsideracompactthreedimensionalpseudo-Hermitianmanifold.M;J;(cid:3)/ (with no boundary) of positive Tanaka–Webster class. This means that the first eigenvalueoftheconformalsublaplacian L WD(cid:2)4(cid:4) CR b b isstrictlypositive.Here,(cid:4) standsforthesublaplacianofM,andRfortheTanaka– b Webster curvature. The conformal sublaplacian rules the change of the Tanaka– Webstercurvatureundertheconformaldeformation(cid:3)O Du2(cid:3) throughthefollowing formula (cid:2)4(cid:4) uCRuDROu3; b where RO is the Tanaka–Webster curvature correspondingto the pseudo-Hermitian structure .J;(cid:3)O/. The positivity of the Tanaka–Webster class is equivalent to the J.-H.Cheng((cid:2)) InstituteofMathematics,AcademiaSinica,Taipei,Taiwan e-mail:[email protected] A.Malchiodi MathematicsInstitute,UniversityofWarwick,Coventry,UK e-mail:[email protected] P.Yang DepartmentofMathematics,PrincetonUniversity,Princeton,NJ,USA e-mail:[email protected] ©SpringerInternationalPublishingSwitzerland2015 3 M.delMarGonzálezetal.(eds.),ExtendedAbstractsFall2013, TrendsinMathematics,DOI10.1007/978-3-319-21284-5_1