Einstein-like geometric structures on surfaces (Q2864585)

In the work [``Geometric structures modeled on affine hypersurfaces and generalizations of the Einstein Weyl and affine hypersphere equations'', Preprint, \url{arXiv:0909.1897}], the author defined a class of geometric structures, called affine hypersurface structures (AH structures) and affine hypersurface equations. They are the generalizations of Weyl structures (equations) etc. In this paper, the author solves these equations on compact orientable surfaces and describes some aspects of their geometry.NEWLINENEWLINEThe contents of this paper is:NEWLINENEWLINE1. IntroductionNEWLINENEWLINE2. Notation and terminologyNEWLINENEWLINE3. Holomorphic differentials and conformal Killing and Codazzi tensorsNEWLINENEWLINE4. AH structures on surfacesNEWLINENEWLINE5. Curvature of an AH structureNEWLINENEWLINE6. Einstein equationsNEWLINENEWLINE7. Classification of Einstein AH structures by scalar curvature and genusNEWLINENEWLINE8. Relation with Abelian vortex equationsNEWLINENEWLINE9. Einstein AH structures on compact orientable surfaces of genus at least twoNEWLINENEWLINE{10}. Einstein-Weyl structures on the sphere and torusNEWLINENEWLINE{11}. Convexity and Hessian metricsNEWLINENEWLINE{12}. Lagrangian immersions in (para)-Kähler space forms