Global Affine Differential Geometry of Hypersurfaces

Polynomial cubic differentials and convex polygons in the projective plane

“…As we noted in the discussion of the completeness of affine spheres in Section 4.1, the seminal existence result for hyperbolic affine spheres is due to ChengYau [CY75], with some clarifications on the notions of completeness (see the last sentence of the statement below) due to Gigena [Gig81], Sasaki [Sas80], and Li [Li90] [Li92]. (The book [LSZ93] gives a comprehensive and coherent account of this theory.) The Cheng-Yau result says that hyperbolic affine spheres of a given mean curvature in R 3 correspond to properly convex sets in RP 2 .…”

Completness of Statistical Structures

“…In this section we introduce notions appearing in this paper and provide basic information on them. All details can be found in [7], [6] and [9].…”

“…There are a few very famous theorems in this respect in the case of Blaschke hypersurfaces, like theorems of Blaschke, Calabi, Cheng-Yau, see e.g. [2], [3], [6]. As concerns the completeness of the induced affine connection (which is usually non-metrizable) it was only noticed by Nomizu, see e.g.…”

The geometric Cauchy problem for the hyperbolic Hessian one equation

“…Consider ψ : Σ −→ R 3 an indefinite improper affine sphere, that is, an immersion with constant affine normal ξ and Lorentzian affine metric h. Then, see [10,17], up to an equiaffine transformation, one has ξ = (0, 0, 1) and ψ can be locally seen as the graph of a solution f (x, y) of (1.1).…”