Singularities of improper affine maps and their Hessian equation

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.

Topology and its Applications, 2012

Generic singularities of envelopes of families of chords and bifurcations of affine equidistants defined by a pair of a curve and a surface in R 3 are classified. The chords join pairs of points of the curve and the surface such that the tangent line to the curve is parallel to the tangent plane to the surface. The classification contains singularities of stable Lagrange and Legendre projections, boundary singularities and some less known classes appearing at the points of the surface and the curve themselves.

Cornell University - arXiv, 2022

We consider in this paper discrete improper affine spheres based on asymptotic nets. In this context, we distinguish the discrete edges and vertices that must be considered singular. The singular edges can be considered as discrete cuspidal edges, while some of the singular vertices can be considered as discrete swallowtails. The classification of singularities of discrete nets is a quite difficult task, and our results can be considered as a fisrt step in this direction. We also prove some characterizations of ruled discrete improper affine spheres which are analogous to the smooth case.

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 Algebra, 2004

Let k be a field of characteristic zero. For small n, we classify all f ∈ k [n] such that the Hessian of f is singular.

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.

Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica

This article presents the theory of focal locus applied to the hypersurfaces in the projective space which are (finitely) covered by linear spaces and such that the tangent space is constant along these spaces.

Journal of Mathematics of Kyoto University, 2008

We consider normal affine surfaces X withétale endomorphisms. It is proved that if X has at least one singular point which is not a quotient singular point then such an endomorphism is an isomorphism.

Grundlehren der mathematischen Wissenschaften, 2020

arXiv (Cornell University), 2018

Given a rank-two sub-Riemannian structure (M, ∆) and a point x 0 ∈ M , a singular curve is a critical point of the endpoint map F : γ → γ(1) defined on the space of horizontal curves starting at x 0 . The typical least degenerate singular curves of these structures are called regular singular curves; they are nice if their endpoint is not conjugate along γ. The main goal of this paper is to show that locally around a nice singular curve γ, once we choose a suitable topology on the control space we can find a normal form for the endpoint map, in which F writes as a sum of a linear map and a quadratic form. We also study the restriction of F to the level sets of the action functional and give a Morse-like formula for the inertia index of its Hessian at γ. This is a preparation for a forthcoming generalization of the Morse theory to rank-two sub-Riemannian structures.

2012

I also acknowledge the generous funding from EPSRC. CONTENTS 4.2. Versality of singularities in the non-transversal setting 4.3. Calculations from the normal forms in the non-transversal setting 5. The Minkowski set in the tangential setting 5.1. Recognition of singularities in the tangential setting 5.2. Versality of singularities in the tangential setting 6. The Minkowski set in the bitangential setting 6.1. Recognition of singularities in the bitangential setting 6.2. Versality of singularities in the bitangential setting 6.3. Wave fronts in the non-transversal, tangential and bitangential settings 7. The Minkowski set in the parallel setting 7.1. Classification 7.2. Recognition of Parallel Case Singularities Chapter 5. Future directions Bibliography CHAPTER 1 Proposition 2.1. The wave front W (F) coincides with the extended affine equidistant W (M, N ) plus two components M and N themselves.

International Journal of Algebra, 2019

We obtain a list of all simple classes of singularities of map germs from R 2 to R 3 with respect to the quasi equivalence relation. A comparison between quasi and Mond's classes and their relations with boundary (Arnold's) and corner classes are discussed.

Results in Mathematics, 2009

The paper deals with the study of affine maximal surfaces with admissible singularities. We shall give some fundamental notions, results, tools and open problems that keep unsolved up to date.

International Journal of Bifurcation and Chaos, 2005

This paper continues the study of the global dynamic properties specific to maps of the plane characterized by the presence of a denominator that vanishes in a one-dimensional submanifold. After two previous papers by the same authors, where the definitions of new kinds of singularities, called focal points and prefocal sets, are given, as well as the particular structures of the basins and the global bifurcations related to the presence of such singularities, this third paper is devoted to the analysis of nonsimple focal points, and the bifurcations associated with them. We prove the existence of a one-to-one relation between the points of a prefocal curve and arcs through the focal point having all the same tangent but different curvatures. In the case of nonsimple focal points, such a relation replaces the one-to-one correspondence between the slopes of arcs through a focal point and the points along the associated prefocal curve that have been proved and extensively discussed in the previous papers. Moreover, when dealing with noninvertible maps, other kinds of relations can be obtained in the presence of nonsimple focal points or prefocal curves, and some of them are associated with qualitative changes of the critical sets, i.e. with the structure of the Riemann foliation of the plane.

2016

I also acknowledge the generous funding from EPSRC. Contents Chapter 1. Introduction Chapter 2. General definitions, the two cases considered and a summary of results 1. General definitions CONTENTS 4.2. Versality of singularities in the non-transversal setting 4.3. Calculations from the normal forms in the non-transversal setting 5. The Minkowski set in the tangential setting 5.1. Recognition of singularities in the tangential setting 5.2. Versality of singularities in the tangential setting 6. The Minkowski set in the bitangential setting 6.1. Recognition of singularities in the bitangential setting 6.2. Versality of singularities in the bitangential setting 6.3. Wave fronts in the non-transversal, tangential and bitangential settings 7. The Minkowski set in the parallel setting 7.1. Classification 7.2. Recognition of Parallel Case Singularities Chapter 5. Future directions Bibliography CHAPTER Proposition 2.1. The wave front W (F) coincides with the extended affine equidistant W (M, N) plus two components M and N themselves.