Ribaucour type transformations for the Hessian one equation

Abstract

We extend the classical theory of Ribaucour transformations to the family of improper affine maps and use it to obtain new solutions of the hessian one equation. We prove that such transformations produce complete, embedded ends of parabolic type and curves of singularities which generically are cuspidal edges. Moreover, we show that these ends and curves of singularities do no intersect. We apply Ribaucour transformations to some helicoidal improper affine maps providing new 3-parameter families with an interesting geometry and a good behavior at infinity. In particular, we construct improper affine maps, periodic in one variable, with any even number of complete embedded ends.2000 Mathematical Subject Classification: 53A15

Improper affine spheres and the Hessian one equation

“…In very recent works, [18,19,20,21] we have solved 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; the problem is well-posed when the initial data are non-characteristic and the uniqueness of the solution can fail at characteristic directions. We also have learnt how to obtain easily improper affine maps with a prescribed singular set, how to construct global examples with the desired singularities and how to use the classical theory of Ribaucour transformations to obtain new solutions of the elliptic hessian one equation.…”