HOLOGRAPHY OF GEODESIC FLOWS, HARMONIZING METRICS, AND BILLIARDS’ DYNAMICS
Gabriel Katz (MIT)
Thursday, March 6, 2025, 11:10 – 12:00, -101
Abstract:
| Let (M,g) be a Riemannian manifold with boundary, where g is a non-trapping metric. Let SM be the space of the spherical tangent to M bundle, and vg the geodesic vector field on SM. We study the scattering maps Cvg : ∂1+SM → ∂1−SM, generated by the vg-flow, and the dynamics of the billiard maps Bvg,τ : ∂1+SM → ∂1+SM, where τ denotes an involution, mimicking the elastic reflection from the the boundary ∂M. We get a variety of holography theorems that tackle the inverse scattering problems for Cvg and theorems that describe the dynamics of Bvg ,τ . Our main tools are a Lyapunov function F : SM → R for vg and a special harmonizing Riemannian metrics g• on SM, a metric in which dF is harmonic. For such metrics g•, we get a family of isoperimetric inequalities of the type volg• (SM) ≤ volg• | (∂(SM)) and for- mulas for the average volume of the minimal hypesufaces {F−1(c)}c∈F(SM). We investigate the interplay between the harmonizing metrics g• and the clas- sical Sasaki metric gg on SM. Assuming ergodicity of Bvg,τ, we also get Santal ́o-Chernov type formulas for the average length of free geodesic segments in M and for the average variation of the Lyapunov function F along the vg-trajectories. |