[Paper Review] Variation of the Liouville measure of a hyperbolic surface

Francis Bonahon, Yaşar Sözen|arXiv (Cornell University)|Mar 11, 2002

Geometric Analysis and Curvature Flows6 references4 citations

TL;DR

This paper establishes the differentiability of the Liouville current map from Teichmüller space to the space of Hölder geodesic currents on a hyperbolic surface. By showing that the derivative of the Liouville measure under metric variation yields a Hölder distribution, it provides a differential calculus for the space of geodesic currents, extending the classical Liouville measure to a differentiable structure in Teichmüller theory.

ABSTRACT

For a compact riemannian manifold of negative curvature, the geodesic foliation of its unit tangent bundle is independent of the negatively curved metric, up to Holder bicontinuous homeomorphism. However, the riemannian metric defines a natural transverse measure to this foliation, the Liouville transverse measure, which does depend on the metric. For a surface S, we show that the map which to a hyperbolic metric on S associates its Liouville transverse measure is differentiable, in an appropriate sense. Its tangent map is valued in the space of transverse Holder distributions for the geodesic foliation.

Motivation & Objective

• To establish the differentiability of the Liouville current map $ L: \mathcal{T}(S) \to \mathcal{C}(S) $, where $ \mathcal{T}(S) $ is the Teichmüller space of a compact hyperbolic surface $ S $.
• To show that the derivative of the Liouville measure under metric variation lies in the space of transverse Hölder distributions for the geodesic foliation.
• To extend the classical Liouville measure to a differentiable structure on Teichmüller space by embedding the tangent space into the space of Hölder geodesic currents.
• To provide a formula for the tangent map $ T_mL $ in terms of the shearing cocycle and geometric data of the universal cover.

Proposed method

• The authors use the geodesic foliation of the unit tangent bundle $ T^1S $, which is independent of the hyperbolic metric up to Hölder bicontinuous homeomorphism.
• They define the Liouville current $ L_m $ as the transverse measure to the geodesic foliation induced by the hyperbolic metric $ m $.
• By analyzing the variation of $ L_m $ along differentiable curves $ m_t $ in $ \mathcal{T}(S) $, they compute the derivative $ \frac{d}{dt}L_{m_t}\big|_{t=0} $ as a limit of measures.
• They show that this derivative defines a Hölder geodesic current by proving convergence of a series involving the shearing cocycle $ \dot{\sigma}_0 $ and geometric coefficients $ C_0(\varphi, T) $.
• The key technical tool is a comparison of distances and angles in the universal cover $ \widetilde{S} $ under varying metrics, using the exponential decay of certain terms.
• They apply the dominated convergence theorem and uniform convergence estimates to justify differentiation under the integral sign.

Experimental results

Research questions

• RQ1 Is the map $ L: \mathcal{T}(S) \to \mathcal{C}(S) $, which assigns to each hyperbolic metric its Liouville current, differentiable?
• RQ2 What is the nature of the derivative of the Liouville current under variation of the hyperbolic metric?
• RQ3 Can the derivative be represented as a Hölder distribution on the space of geodesics in the universal cover?
• RQ4 How does the tangent map $ T_mL $ relate to the shearing cocycle of the geodesic lamination?
• RQ5 Is the tangent map $ T_mL $ continuous in the base metric $ m \in \mathcal{T}(S) $?

Key findings

• The Liouville current map $ L: \mathcal{T}(S) \to \mathcal{C}(S) $ is differentiable at every point $ m \in \mathcal{T}(S) $, with the derivative valued in the space of Hölder geodesic currents $ \mathcal{H}(S) $.
• The tangent map $ T_mL: T_m\mathcal{T}(S) \to \mathcal{H}(S) $ is linear and depends continuously on $ m $.
• The derivative $ T_mL(\dot{m}) $ is given explicitly by the formula $ T_mL(\dot{m})(\varphi) = \sum_T \dot{\sigma}_0(T) C_0(\varphi, T) $, where $ \dot{\sigma}_0 $ is the shearing cocycle and $ C_0(\varphi, T) $ are geometric coefficients.
• The convergence of the series defining the derivative is uniform on compact subsets of $ \mathcal{T}(S) $, ensuring continuity of the tangent map.
• The derivative is invariant under the action of $ \pi_1(S) $, confirming that it defines a well-defined Hölder geodesic current on the surface.
• The result extends the classical Liouville measure to a differentiable structure on Teichmüller space, enabling differential geometry on the space of geodesic currents.