Abstract



An AH (affine hypersurface) structure is a pair comprising a projective equivalence class of torsionfree 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 cooriented nondegenerate 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 MongeAmp\`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 paraK\"ahler fourmanifold with constant paraholomorphic 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 PisaClasse Di Scienze 
ISBN

0391173X 
Impact factor JCR

0,906 
Impact info

JCR 2013. 54 de 299 en matematicas. Q1. 
Volume

XII 

10.2422/20362145.201101_002 
Journal number

3 
From page

499 
To page

585 
Month

SIN MES 
Ranking

JCR 2013. 54 de 299 en matematicas. Q1. 