Quasi-Fuchsian flows and the coupled vortex equations

TL;DR

This paper introduces a new construction of quasi-Fuchsian flows using coupled vortex equations, viewing them as thermostats on the tangent bundle of the Blaschke metric, and provides formulas for their marked length spectrum.

Contribution

It offers an alternative approach to quasi-Fuchsian flows through coupled vortex equations, connecting them with thermostat models and explicit spectral formulas.

Findings

New construction of quasi-Fuchsian flows via coupled vortex equations

Representation of flows as thermostats on the tangent bundle

Formulas for the marked length spectrum in thermostat parametrization

Abstract

We provide an alternative construction of the quasi-Fuchsian flows introduced by Ghys in \cite{Ghys-92}. Our approach is based on the coupled vortex equations that allows to see these flows as thermostats on the unit tangent bundle of the Blaschke metric uniquely determined by a conformal class and a holomorphic quadratic differential. We also give formulas for the marked length spectrum of a quasi-Fuchsian flow in the thermostat parametrization.