Ribaucour type transformations for the Hessian one equation. (January 2015)

• Record Type: Journal Article
• Authors: Martínez, Antonio; Milán, Francisco; Tenenblat, Keti
• Is Part Of: Nonlinear analysis. Volume 112(2015)
• Page Start: 147
• Page End: 155
• Publication Date: 2015-01
• Subjects: 53A15; Ribaucour transformations -- Improper affine spheres -- Hessian one equation; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2014.09.013
• Languages: English
• ISSNs: 0362-546X

Abstract

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.