Resumen de Einstein-like geometric structures on surfaces
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.
