Some Geometric Aspects of the Hessian One Equation

Differential Geometry and its Applications, 2017

Improper affine spheres have played an important role in the development of geometric methods for the study of the Hessian one equation. Here, we review most of the advances we have made in this direction during the last twenty years.

Advances in Mathematics, 2014

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.

Journal of Mathematical Analysis and Applications, 2013

We study improper affine spheres with some admissible singularities, called improper affine maps and associated to the unimodular Hessian equation. In particular, we characterize when a curve of R 3 is the singular curve of some improper affine map with prescribed cuspidal edges and swallowtails. Also, we consider improper affine maps with isolated singularities and show some similarities and differences between the Hessian +1 equation and the Hessian −1 equation. As a consequence, we construct global examples with the desired singularities.

Nonlinear Analysis: Theory, Methods & Applications, 2015

We solve the problem of finding all indefinite improper affine spheres passing through a given regular curve of R 3 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.

Journal of Mathematical Analysis and Applications, 2015

There are exactly two different types of bi-dimensional improper affine spheres: the non-convex ones can be modeled by the centerchord transform of a pair of planar curves while the convex ones can be modeled by a holomorphic map. In this paper, we show that both constructions can be generalized to arbitrary even dimensions: the former class corresponds to the center-chord transform of a pair of Lagrangian submanifolds while the latter is related to special Kähler manifolds. Furthermore, we show that the improper affine spheres obtained in this way are solutions of certain exterior differential systems. Finally, we also discuss the problem of realization of simple stable Legendrian singularities as singularities of these improper affine spheres.

Journal of Geometry, 2011

Given a pair of planar curves, one can define its generalized area distance, a concept that generalizes the area distance of a single curve. In this paper, we show that the generalized area distance of a pair of planar curves is an improper indefinite affine spheres with singularities, and, reciprocally, every indefinite improper affine sphere in R 3 is the generalized distance of a pair of planar curves. Considering this representation, the singularity set of the improper affine sphere corresponds to the area evolute of the pair of curves, and this fact allows us to describe a clear geometric picture of the former. Other symmetry sets of the pair of curves, like the affine area symmetry set and the affine envelope symmetry set can be also used to describe geometric properties of the improper affine sphere.

Advances in Mathematics, 2020

Given a Lagrangian submanifold L of the affine symplectic 2n-space, one can canonically and uniquely define a center-chord and a special improper affine sphere of dimension 2n, both of whose sets of singularities contain L. Although these improper affine spheres (IAS) always present other singularities away from L (the off-shell singularities studied in [6]), they may also present singularities other than L which are arbitrarily close to L, the so called singularities "on shell". These on-shell singularities possess a hidden Z2 symmetry that is absent from the offshell singularities. In this paper, we study these canonical IAS obtained from L and their on-shell singularities, in arbitrary even dimensions, and classify all stable Lagrangian/Legendrian singularities on shell that may occur for these IAS when L is a curve or a Lagrangian surface.

Journal de Mathématiques Pures et …, 2005

We construct the space of solutions to the elliptic Monge-Ampère equation det(D 2 φ) = 1 in the plane R 2 with n points removed. We show that, modulo equiaffine transformations and for n > 1, this space can be seen as an open subset of R 3n−4 , where the coordinates are described by the conformal equivalence classes of once punctured bounded domains in C of connectivity n − 1. This approach actually provides a constructive procedure that recovers all such solutions to the Monge-Ampère equation, and generalizes a theorem by K. Jörgens.

2022

In this note we discuss the geometry of Riemannian surfaces having a discrete set of singular points. We assume the conformal structure extends through the singularities and the curvature is integrable. Such points are called \emph{simple singularities}. We first describe them locally and then globally using the notion of (real) divisor. We formulate a Gauss-Bonnet formula and relate it to some asymptotic isoperimetric ratio. We prove a classifications theorem for flat metrics with simple singularities on a compact surface and discuss the Berger--Nirenberg Problem on surfaces with a divisor. We finally discuss the relation with spherical polyhedra.

Mathematische Annalen, 1994