Tangent ray foliations and outer billiards

Início: 14/06/2024 17:00

Término: 14/06/2024 18:00

Palestrante: Yamile Godoy (Universidad Nacional de Córdoba)

E-mail do Palestrante: yamile.godoy@unc.edu.ar

Resumo: Given a smooth closed strictly convex curve $gamma$ in the plane and a point $x$ outside of $gamma$, there are two tangent lines to $gamma$ through $x$; choose one of them consistently, say, the right one from the viewpoint of $x$, and the outer billiard map $B$ is defined by reflecting $x$ about the point of tangency. We observe that the good definition and the injectivity of the plane outer billiard map is a consequence of the fact that the tangent rays associated to both tangent vectors to $gamma$ determine foliations of the exterior of the curve. In this talk, we will present the results obtained from a generalization of the problem of defining outer billiards in higher dimensions. Let $v$ be a smooth unit vector field on a complete, umbilic (but not totally geodesic) hypersurface $N$ in a space form; for example on the unit sphere $S^{2k-1} subset mathbb{R}^{2k}$, or on a horosphere in hyperbolic space. We give necessary and sufficient conditions on $v$ for the rays with initial velocities $v$ (and $-v$) to foliate the exterior $U$ of $N$. We find and explore relationships among these vector fields and geodesic vector fields on $N$. When the rays corresponding to each of $pm v$ foliate $U$, $v$ induces an outer billiard map whose billiard table is $U$. We describe the unit vector fields on $N$ whose associated outer billiard map is volume preserving. This is a joint work with Michael Harrison (Institute for Advanced Study, Princeton) and Marcos Salvai (UNC, Argentina).