Einstein-like geometric structures on surfaces

Fox, Daniel Jeremy Forrest (2013). Einstein-like geometric structures on surfaces. "Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze", v. XII (n. 3); pp. 499-585. ISSN 0391-173X. https://doi.org/10.2422/2036-2145.201101_002.


Título: Einstein-like geometric structures on surfaces
  • Fox, Daniel Jeremy Forrest
Tipo de Documento: Artículo
Título de Revista/Publicación: Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze
Fecha: 2013
Volumen: XII
Escuela: E.U.I.T. Industrial (UPM) [antigua denominación]
Departamento: Matemática Aplicada
Licencias Creative Commons: Reconocimiento - Sin obra derivada - No comercial

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An AH (affine hypersurface) structure is a pair comprising a projective equivalence class of torsion-free connections and a conformal structure satisfying a compatibility condition which is automatic in two dimensions. They generalize Weyl structures, and a pair of AH structures is induced on a co-oriented non-degenerate immersed hypersurface in flat affine space. The author has defined for AH structures Einstein equations, which specialize on the one hand to the usual Einstein Weyl equations and, on the other hand, to the equations for affine hyperspheres. Here these equations are solved for Riemannian signature AH structures on compact orientable surfaces, the deformation spaces of solutions are described, and some aspects of the geometry of these structures are related. Every such structure is either Einstein Weyl (in the sense defined for surfaces by Calderbank) or is determined by a pair comprising a conformal structure and a cubic holomorphic 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 appropriate power of the canonical bundle. On the cone over a surface of genus at least two carrying an Einstein AH structure there are Monge-Amp`ere metrics of Lorentzian and Riemannian signature and a Riemannian Einstein K"ahler affine metric. A mean curvature zero spacelike immersed Lagrangian submanifold of a para-K"ahler four-manifold with constant para-holomorphic sectional curvature inherits an Einstein AH structure, and this is used to deduce some restrictions on such immersions.

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ID de Registro: 33187
Identificador DC: http://oa.upm.es/33187/
Identificador OAI: oai:oa.upm.es:33187
Identificador DOI: 10.2422/2036-2145.201101_002
URL Oficial: http://annaliscienze.sns.it/index.php?page=Article&id=276
Depositado por: Memoria Investigacion
Depositado el: 30 Ene 2015 09:40
Ultima Modificación: 06 Feb 2015 11:34
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