# Hamiltonian properties of earthquakes flows on surfaces with closed geodesic boundary

## TL;DR

This paper demonstrates that earthquake flows on hyperbolic surfaces with boundary are Hamiltonian, extending classical length functions, and applies this to classify certain affine representations of surface groups.

## Contribution

It establishes the Hamiltonian nature of earthquake flows on bordered surfaces and introduces a new Hamiltonian function extending classical length maps.

## Key findings

- Earthquake flows are Hamiltonian on Teichmüller space with fixed boundary lengths.
- The Hamiltonian function extends classical length maps and is proper and strictly convex.
- Application to classifying affine representations with fixed boundary lengths.

## Abstract

The Teichm\"uller space $\mathcal{T}_S(\mathbf{b})$ of hyperbolic metrics on a surface $S$ with fixed lengths at the boundary components is symplectic. We prove that any sum of infinitesimal earthquakes on $S$ that is tangent to $\mathcal{T}_S(\mathbf{b})$ is Hamiltonian, by providing a Hamiltonian $\mathbb{L}$. Such function extends the classical length map associated to a compactly supported measured geodesic lamination and shares with it some peculiar properties, such as properness and strict convexity along earthquakes paths under usual topological conditions. As an application, we prove that any non-Fuchsian affine representation of $\pi_1(S)$ into $\mathbb{R}^{2,1}\rtimes SO_0(2,1)$ with cocompact discrete linear part is determined by the singularities of the two invariant regular domains in $\mathbb{R}^{2,1}$ pointed out by Barbot, once the boundary lengths are fixed.