Even Dimensional Improper Affine Spheres

Abstract

There are exactly two different types of bi-dimensional improper affine spheres: the non-convex ones can be modeled by the center-chord 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.

26 pages

References in corpus (3)

• The Wigner caustic on shell and singularities of odd functions
• Survey on Affine Spheres
• Singularities of equidistants and global centre symmetry sets of Lagrangian submanifolds

Cited by in corpus (7)

• The improved isoperimetric inequality and the Wigner caustic of planar ovals
• Singular points of the Wigner caustic and affine equidistants of planar curves
• The Gauss-Bonnet Theorem for coherent tangent bundles over surfaces with boundary and its applications
• Singular Improper Affine Spheres from a given Lagrangian Submanifold
• Isoperimetric equalities for rosettes
• Singularities of affine equidistants: extrinsic geometry of surfaces in 4-space
• Equiaffine Characterization Of Lagrangian Surfaces in $R^4$