Pedal Curves of Non-Lightlike Curves in Minkowski 3-Space

We study the notions of pedal curves, contrapedal curves and B-Gauss maps of non-lightlike regular curves in Minkowski 3-space. Then we establish the relationships among the evolutes, the pedal and contrapedal curves. Moreover, we also investigate the singularities of these objects. Finally, we show some examples to comprehend the characteristics of the pedal and contrapedal curves in Minkowski 3-space.

Semicontinuity of Gauss maps and the Schottky problem

We show that the degree of Gauss maps on abelian varieties is semicontinuous in families, and we study its jump loci. As an application we obtain that in the case of theta divisors this degree answers the Schottky problem. Our proof computes the degree of Gauss maps by specialization of Lagrangian cycles on the cotangent bundle. We also get similar results for the intersection cohomology of varieties with a finite morphism to an abelian variety; it follows that many components of Andreotti–Mayer loci, including the Schottky locus, are part of the stratification of the moduli space of ppav’s defined by the topological type of the theta divisor.

Gauss Maps of the Surfaces in the Three-Dimensional Heisenberg Group

In this paper we study the Gauss map of surfaces in three-dimensional Heisenberg group using the Gans model of the hyperbolic plane. We establish a relationship between the tension fields of the Gauss maps and the mean curvatures of the surfaces in $\mathcal{H}_{3}$.