Francisco Milán

An extension of a theorem by K. Jörgens and a maximum principle at infinity for parabolic affine spheres

where Ω is a planar domain and f is in the usual Holder space C2,α(Ω). Without loss of generality we shall consider only locally convex solutions of (1). This equation arises in the context of an affine differential geometric problem as the equation of a parabolic affine sphere (in short PA-sphere) in the unimodular affine real 3-space (see [C1], [C2], [CY] and [LSZ]). Contrary to the case of smooth bounded convex domains, little is known about solutions of (1) when the domain is unbounded. Here, we recall a famous result by K. Jorgens which asserts that any solution of (1) on Ω = R2 is a quadratic polynomial (see [J]) and we also mention a previous paper (see [FMM]) where the authors study solutions of (1) on the exterior of a planar domain that are regular at infinity. Since the underlying almost-complex structure of (1) is integrable, one expects PA-spheres (with their canonical conformal structure) to be conveniently described in terms of meromorphic functions. The reader will find in Sect. 2 a complex representation of PA-spheres and, particularly, a complex description for the solutions of (1).

Complete linear Weingarten surfaces of Bryant type. A Plateau problem at infinity

In this paper we study a large class of Weingarten surfaces which includes the constant mean curvature one surfaces and flat surfaces in the hyperbolic 3-space. We show that these surfaces can be parametrized by holomorphic data like minimal surfaces in the Euclidean 3-space and we use it to study their completeness. We also establish some existence and uniqueness theorems by studing the Plateau problem at infinity: when is a given curve on the ideal boundary the asymptotic boundary of a complete surface in our family? and, how many embedded solutions are there?

Symmetry and uniqueness of parabolic affine spheres

The proof of a variety of problems about symmetry properties in partial differential equations and differential geometry is based on the Maximum Principle for elliptic linear equations (see [A1], [GNN] and [S]). In this paper we apply a similar technique to a locally strongly convex parabolic affine sphere (in short, PA-sphere) embedded in the unimodular affine real 3-space j~3. The study of PA-spheres is locally equivalent (see [C1], [C2]) to the study of convex solutions of the Monge-Amp~re equation

Flat Surfaces in \(\mathbb{L}\)4

AbstractWe give a global conformal representation for flat surfaces with a flat normal bundle in the standard flat Lorentzian space form

Pick invariant and affine Gauss-Kronecker curvature

\mathbb{L}

Improper affine spheres and the Hessian one equation

4. Particularly, flat surfaces in hyperbolic 3-space, the de Sitter 3-space, the null cone, and other numerous examples aredescribed.

Some Geometric Aspects of the Hessian One Equation

In this paper we classify in a global way the umbilical affine definite surfaces in R4 with respect to the Nomizu-Vrancken affine normalization introduced in [NV]. We prove that an affine complete affine definite surface in R4 is umbilical if and only if it is affine equivalent to the complex paraboloid.

Contact Holomorphic Curves and Flat Surfaces

The elliptic paraboloid and the homogeneous affine surface given by (u, v, 1/2(u2+v−2/3)),v>0, are characterized as locally strongly convex affine surfaces inA, with constant Pick invariant and vanishing affine Gauss-Kronecker curvature.