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Quasi-Fuchsian flows and the coupled vortex equations

Mihajlo Cekić, Gabriel P. Paternain

TL;DR

This work shows that Ghys's quasi-Fuchsian flows can be realized explicitly as thermostat flows on the unit tangent bundle, driven by a Blaschke metric in a fixed conformal class together with a holomorphic quadratic differential $A$ of degree $2$ via the coupled vortex equations. A central construction is the bijection $\mathcal{G}: \mathcal{T}\mathrm{QD}(M)\to \mathcal{T}(M)\times\mathcal{T}(M)$ built from harmonic maps and their Hopf differentials, which aligns the weak foliations with those of two hyperbolic metrics $g_1,g_2$. The main theorem provides a precise PDE-based correspondence: explicit foliation conjugacies, a volume-preservation condition equivalent to $[g_1]_{Teich}=[g_2]_{Teich}$ (equivalently $A=0$), and a marked length spectrum relation where the closed orbit length equals the arithmetic mean of the corresponding hyperbolic lengths. The MLS identities yield a PDE-based route to hyperbolic MLS rigidity and illuminate how the thermostat data encode the quasi-Fuchsian pair; the results complement Ghys's geometric construction by tying the flows to Dirichlet-energy minimization and harmonic map theory.

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.

Quasi-Fuchsian flows and the coupled vortex equations

TL;DR

This work shows that Ghys's quasi-Fuchsian flows can be realized explicitly as thermostat flows on the unit tangent bundle, driven by a Blaschke metric in a fixed conformal class together with a holomorphic quadratic differential of degree via the coupled vortex equations. A central construction is the bijection built from harmonic maps and their Hopf differentials, which aligns the weak foliations with those of two hyperbolic metrics . The main theorem provides a precise PDE-based correspondence: explicit foliation conjugacies, a volume-preservation condition equivalent to (equivalently ), and a marked length spectrum relation where the closed orbit length equals the arithmetic mean of the corresponding hyperbolic lengths. The MLS identities yield a PDE-based route to hyperbolic MLS rigidity and illuminate how the thermostat data encode the quasi-Fuchsian pair; the results complement Ghys's geometric construction by tying the flows to Dirichlet-energy minimization and harmonic map theory.

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.
Paper Structure (11 sections, 6 theorems, 76 equations)

This paper contains 11 sections, 6 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Geometry of the unit circle bundle
  4. Coupled vortex equations
  5. Harmonic maps and the Hopf differential
  6. The bijection $\mathcal{G}$
  7. The conjugacy $\mathcal{I}$ and proof of Theorem \ref{['thm:main']}
  8. Proof of Theorem \ref{['thm:main']}
  9. The marked length spectrum of the pair $([g_{1}]_{\mathrm{Teich}}, [g_{2}]_{\mathrm{Teich}})$.
  10. Proof of \ref{['eq:mls1']} and \ref{['eq:mls2']}
  11. Equality of marked length spectra and Proof of \ref{['eq:arithmetic-mean']}

Key Result

Theorem 1.1

The map $\mathcal{G}$ defined by where $[\sigma]_{\mathrm{conf}}$ is a conformal class on $M$ and $A \in \mathrm{QD}([g]_{\mathrm{conf}})$, descends to a bijection $\mathcal{G}: {\mathcal{T}} \mathrm{QD}(M)\to {\mathcal{T}}(M)\times {\mathcal{T}}(M)$ such that if we define $([g_{1}]_{\mathrm{Teich}}, [g_{2}]_{\mathrm{Teich}}) := \m

Theorems & Definitions (14)

  • Theorem 1.1
  • Proposition 3.1
  • proof
  • Remark 3.2
  • Lemma 4.1
  • proof
  • Remark 4.2
  • Lemma 4.3
  • proof
  • Lemma 4.4
  • ...and 4 more