Equiaffine geometry of level sets and ruled hypersurfaces with equiaffine mean curvature zero

TL;DR

This paper systematically develops the equiaffine geometry of level sets and constructs families of high-dimensional ruled hypersurfaces with zero equiaffine mean curvature that solve the affine normal flow.

Contribution

It introduces a systematic approach to equiaffine geometry of level sets and constructs new examples of hypersurfaces with special curvature properties.

Findings

Constructed families of 2n-dimensional hypersurfaces with zero equiaffine mean curvature.

Hypersurfaces are ruled by n-planes and solve the affine normal flow.

Each hypersurface admits a symplectic structure with Lagrangian rulings.

Abstract

Basic aspects of the equiaffine geometry of level sets are developed systematically. As an application there are constructed families of $2n$-dimensional nondegenerate hypersurfaces ruled by $n$-planes, having equiaffine mean curvature zero, and solving the affine normal flow. Each carries a symplectic structure with respect to which the ruling is Lagrangian.