Einstein-like geometric structures on surfaces

TL;DR

This paper explores Einstein-like geometric structures on surfaces, classifies solutions, and relates them to convex projective structures, vortex equations, and special submanifold geometries, revealing new links between affine, conformal, and complex geometries.

Contribution

It classifies Einstein AH structures on surfaces, linking them to convex projective structures and vortex equations, and explores their geometric properties and applications.

Findings

Every Einstein AH structure on a surface is either Einstein Weyl or determined by a convex projective structure.

Solutions are classified and related to vortex equations on line bundles.

Existence of Monge-Ampère metrics and Einstein Kähler affine metrics on cones over surfaces.

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.