Smooth Hamilton-Jacobi solutions for the Horocycle flow

TL;DR

This paper explicitly characterizes all smooth solutions to the Hamilton-Jacobi equation for the horocycle flow, revealing the structure of these solutions in the context of hyperbolic geometry and magnetic Lagrangians.

Contribution

It provides a complete classification of smooth Hamilton-Jacobi solutions for the horocycle flow, a problem previously unresolved in this setting.

Findings

All smooth solutions are explicitly computed.

Solutions correspond to Lagrangian graphs in the energy level set.

The method involves analyzing Lagrangian graphs within the Hamiltonian level set.

Abstract

In this paper we compute all the smooth solutions to the Hamilton-Jacobi equation associated with the horocycle flow. This can be seen as the Euler-Lagrange flow (restricted to the energy level set $E^{-1}(\frac 12)$) defined by the Tonelli Lagrangian $L:T\mathbb H\rightarrow \mathbb R$ given by (hyperbolic) kinetic energy plus the standard magnetic potential. The method we use is to look at Lagrangian graphs that are contained in the level set $\{H=\frac 12\}$, where $H:T^*\mathbb H\rightarrow \mathbb R$ denotes the Hamiltonian dual to $L$.