Teich Webminar – Friday, January 28th,

Sergei Tabachnikov suggested a generalization of Birkhoff’s Conjecture to projective billiards, which also involves its versions on surfaces with not zero constant curvature: spherical and hyperbolic plane. It is worded in dual terms. In particular, consider a closed planar convex C-curve with a closed curved lamination (where the C-curve is a sheet) on the outer side. For any straight line L tangent to curve C at a point P, consider the involution germ of the straight line L by P swapping its two points of intersection with each individual leaf. Suppose that this last involution is a projective transformation of the line L for any point P.

Tabashnikov’s Conjecture states that then the curve C is an ellipse, and the leaves of the lamination form a brush of conics.
In the presentation, we will demonstrate the rational version: under the additional hypothesis, that the lamination in question admits a first rational integral.
We will demonstrate an analogous result in the case where the curve C is a real or complex curve germ and we have a laminating germ as above, having a first rational integral. In this general case, the C curve is always a conical curve. But the leaves of the lamination can be algebraic curves of higher degree. We will present the complete classification of these slips with projective transformation.