Translation surface

Abstract: We will deal with the translation surfaces which are the shapes generated by translating one curve along another one. We focus on the geometry of translation surfaces generated by two algebraic curves in space and study their properties, especially those useful for geometric modelling purposes. It is a classical result that each minimal surface may be obtained as a translation surface generated by an isotropic curve and its complex conjugate. Thus, we can study the minimal surfaces as special instances of translation surfaces. All the results about translation surfaces will be directly applied also to minimal surfaces. Finally, we present a construction of rational isotropic curves with a prescribed tangent field which leads to the description of all rational minimal surfaces. A close relation to surfaces with Pythagorean normals will be also discussed.

Abstract: Publisher Summary The theory of mathematical billiards can be partitioned into three areas: convex billiards with smooth boundaries, billiards in polygons (and polyhedra), and dispersing and semi-dispersing billiards (similarly to differential geometry in which the cases of positive, zero, and negative curvature are significantly different). These areas differ by the types of results and the methods of study. This chapter illustrates the unfolding procedure in the simplest example of a rational polygon, the square. The chapter examines certain examples of Teichmtiller discs that arise in the so-called Veech billiards and their generalizations. Some results on ergodicity of vertical foliations of quadratic differentials are described in the chapter. The chapter presents the constructive proofs of other results on polygonal billiards and flat surfaces. In particular, it gives a new proof of the quadratic upper bound on the number of saddle connections.

Abstract: Various problems of geometry, topology and dynamical systems on surfaces as well as some questions concerning one-dimensional dynamical systems lead to the study of closed surfaces endowed with a flat metric with several cone-type singularities Such flat surfaces are naturally organized into families which appear to be isomorphic to the moduli spaces of holomorphic one-forms One can obtain much information about the geometry and dynamics of an individual flat surface by studying both its orbit under the Teichmuller geodesic flow and under the linear group action In particular, the Teichmuller geodesic flow plays the role of a time acceleration machine (renormalization procedure) which allows to study the asymptotic behavior of interval exchange transformations and of surface foliations This long survey is an attempt to present some selected ideas, concepts and facts in Teichmuller dynamics in a playful way

Abstract: We prove the Zorich-Kontsevich conjecture that the non-trivial Lyapunov exponents of the Teichm\"uller flow on (any connected component of a stratum of) the moduli space of Abelian differentials on compact Riemann surfaces are all distinct. By previous work of Zorich and Kontsevich, this implies the existence of the complete asymptotic Lagrangian flag describing the behavior in homology of the vertical foliation in a typical translation surface.