Einstein-like geometric structures on surfaces

• Daniel J.F. Fox ^[1]
  1. [1] Universidad Politécnica de Madrid

     Universidad Politécnica de Madrid

     Madrid, España

     
• Localización: Annali della Scuola Normale Superiore di Pisa. Classe di scienze, ISSN 0391-173X, Vol. 12, Nº 3, 2013, págs. 499-585
• Idioma: inglés
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• Resumen

  • AnAH(affinehypersurface)structureisapaircomprisingaprojective equivalenceclassoftorsion-freeconnectionsandaconformalstructuresatisfying acompatibilityconditionwhichisautomaticintwodimensions.Theygeneralize Weyl structures, and a pair of AH structures is induced on a co-oriented nondegenerateimmersedhypersurfaceinflataffinespace.Theauthorhasdefinedfor AHstructuresEinsteinequations,whichspecializeontheonehandtotheusual Einstein-Weylequationsand,ontheotherhand,totheequationsforaffinehyperspheres.HeretheseequationsaresolvedforRiemanniansignatureAHstructures oncompactorientablesurfaces,thedeformationspacesofsolutionsaredescribed, andsomeaspectsofthegeometryofthesestructuresarerelated.EverysuchstructureiseitherEinstein-Weyl(inthesensedefinedforsurfacesbyCalderbank)or isdeterminedbyapaircomprisingaconformalstructureandacubicholomorphic differential, and so by a convex flat real projective structure. In the latter case it can be identified with a solution of the Abelian vortex equations on an appropriatepowerofthecanonicalbundle. Ontheconeoverasurfaceofgenus atleasttwocarryinganEinsteinAHstructurethereareMonge-Amp` eremetrics ofLorentzianandRiemanniansignatureandaRiemannianEinsteinK¨ ahleraffine metric. Ameancurvaturezerospace-likeimmersedLagrangiansubmanifoldof apara-K¨ ahlerfour-manifoldwithconstantpara-holomorphicsectionalcurvature inheritsanEinsteinAHstructure,andthisisusedtodeducesomerestrictionson suchimmersions.