Observatorio de I+D+i UPM

Memorias de investigación
Research Publications in journals:
Einstein-like geometric structures on surfaces
Year:2013
Research Areas
  • Differential geometry
Information
Abstract
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.
International
Si
JCR
Si
Title
Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze
ISBN
0391-173X
Impact factor JCR
0,906
Impact info
JCR 2013. 54 de 299 en matematicas. Q1.
Volume
XII
10.2422/2036-2145.201101_002
Journal number
3
From page
499
To page
585
Month
SIN MES
Ranking
JCR 2013. 54 de 299 en matematicas. Q1.
Participants
  • Autor: Daniel Jeremy Fox Hornig (UPM)
Research Group, Departaments and Institutes related
  • Creador: Departamento: Matemática Aplicada (E.U.I.T. Industrial)
S2i 2020 Observatorio de investigación @ UPM con la colaboración del Consejo Social UPM
Cofinanciación del MINECO en el marco del Programa INNCIDE 2011 (OTR-2011-0236)
Cofinanciación del MINECO en el marco del Programa INNPACTO (IPT-020000-2010-22)