Singularities of improper affine spheres and surfaces of constant Gaussian curvature

2005/02/08 by Go-o Ishikawa, Ishikawa, Go-o, Yoshinori Machida +1
Mathematics · #53A15 #53C42 #Differential Geometry (math.DG) #FOS: Mathematics #math.DG #msc:53A15 #msc:53C42

paper · pdf · doi:10.48550/arxiv.math/0502154

20 pages, 1 figures

arxiv created 2005/02/08 · arxiv updated 2009/12/01

Abstract

We study the equation for improper (parabolic) affine spheres from the view point of contact geometry and provide the generic classification of singularities appearing in geometric solutions to the equation as well as their duals. We also show the results for surfaces of constant Gaussian curvatureand for developable surfaces. In particular we confirm that generic singularities appearing in such a surface are just cuspidal edges and swallowtails.

Related

• The classification of complete improper affine spheres with singularities of low total curvature and new examples
• The Cauchy problem for indefinite improper affine spheres and their Hessian equation
• Characterizing singularities of a surface in Lie sphere geometry
• A Geometric Representation of Improper Indefinite Affine Spheres with Singularities
• Families of spherical surfaces and harmonic maps