Smooth Hamilton-Jacobi solutions for the Horocycle flow

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 $E^{-1}(\frac{1}{2})$) defined by the Tonelli Lagrangian $L:T\mathbb{H}\to \mathbb{R}$ 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 $\{H=\frac{1}{2}\}$, where $H:T^{\ast }\mathbb{H}\to \mathbb{R}$ denotes the Hamiltonian dual to $L$.

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