The Cauchy problem for indefinite improper affine spheres and their Hessian equation

We give a conformal representation for indefinite improper affine spheres which solve the Cauchy problem for their Hessian equation. As consequences, we can characterize their geodesics and obtain a generalized symmetry principle. Then, we classify the helicoidal indefinite improper affine spheres and find a new family with complete non flat affine metric. Moreover, we present interesting examples with singular curves and isolated singularities

The geometric Cauchy problem for the hyperbolic Hessian one equation

We solve the problem of finding all indefinite improper affine spheres passing through a given regular curve of R3R3 with a prescribed affine co-normal vector field along this curve. We prove the problem is well-posed when the initial data are non-characteristic and show that uniqueness of the solution can fail at characteristic directions. As application we classify the indefinite improper affine spheres admitting a geodesic planar curve.Ministerio de Educación y Ciencia Grant No. MTM2013-43970-P and Junta de Andalucía Grant No. FQM-325

Representation formula for discrete indefinite affine spheres

We present a representation formula for discrete indefinite affine spheres via loop group factorizations. This formula is derived from the Birkhoff decomposition of loop groups associated with discrete indefinite affine spheres. In particular we show that a discrete indefinite improper affine sphere can be constructed from two discrete plane curves.Comment: 38 page