[Paper Review] Velling-Kirillov metric on the universal Teichmuller curve

Lee-Peng Teo|ArXiv.org|Jun 19, 2002

Geometry and complex manifolds8 references7 citations

TL;DR

This paper introduces the Velling-Kirillov metric, a Hermitian Kähler metric on the universal Teichmüller curve defined via the second variation of spherical areas of deformed domains. It proves that the vertical integration of the square of this metric's symplectic form yields the Weil-Petersson symplectic form on both the universal and finite-dimensional Teichmüller spaces.

ABSTRACT

We extend Velling's approach and prove that the second variation of the spherical areas of a family of domains defines a Hermitian metric on the universal Teichmuller curve, whose pull back to Diff+(S^1)/S^1 coincides with the Kirillov metric. We show that the vertical integration of the square of the symplectic form of Velling-Kirillov metric on the universal Teichmuller curve is the symplectic form that defines the Weil-Petersson metric on the universal Teichmuller space. Restricted to a finite dimensional Teichmuller space, the vertical integration of the corresponding form on the Teichmuller curve is also the symplectic form that defines the Weil-Petersson metric on the Teichmuller space.

Motivation & Objective

• To define a natural Hermitian Kähler metric on the universal Teichmüller curve using second variations of spherical areas.
• To establish a connection between Velling's area-based metric and Kirillov's coadjoint orbit metric on Diff+(S¹)/S¹.
• To show that vertical integration of the square of the Velling-Kirillov metric's symplectic form recovers the Weil-Petersson metric on Teichmüller spaces.
• To provide a new geometric realization of the Weil-Petersson metric via integration over fibers of the universal Teichmüller curve.

Proposed method

• Define the Velling-Kirillov metric via the second derivative of spherical area of domains f^{tQ}(Δ) under Schwarzian flow.
• Use the Bers embedding of the universal Teichmüller space into A∞(Δ) to identify tangent vectors with holomorphic quadratic differentials.
• Construct the metric on the universal Teichmüller curve by right-translation invariance from the origin, using univalent functions in D̃.
• Prove the metric is Kähler and coincides with Kirillov's metric on Diff+(S¹)/S¹ via vector field expansion ∑n|cn|².
• Define the symplectic form κ of the Velling-Kirillov metric and perform vertical integration over fibers of p: T(1) → T(1).
• Show that the resulting (1,1)-form ω on T(1) is the Weil-Petersson symplectic form, restricted to H^{3/2} vector fields.

Experimental results

Research questions

• RQ1 Can Velling’s area-based metric on the universal Teichmüller space be extended to a natural metric on the universal Teichmüller curve?
• RQ2 Does the pullback of the Velling-Kirillov metric to Diff+(S¹)/S¹ coincide with Kirillov’s coadjoint orbit metric?
• RQ3 Is the vertical integration of the square of the Velling-Kirillov metric’s symplectic form equivalent to the Weil-Petersson metric on T(1)?
• RQ4 Does this construction restrict correctly to finite-dimensional Teichmüller spaces associated with cofinite Fuchsian groups?

Key findings

• The Velling-Kirillov metric is a unique right-invariant Hermitian Kähler metric on the universal Teichmüller curve.
• The metric’s explicit formula is ‖Q‖_S² = ∑_{n=2}^∞ n|a_n|² for Q(z) = ∑_{n=2}^∞ (n³−n)a_n z^{n−2} in A∞(Δ).
• The pullback of the Velling-Kirillov metric to Diff+(S¹)/S¹ matches Kirillov’s metric ∑_{n=1}^∞ n|c_n|² for vector fields ∑c_n e^{inθ} ∂_θ.
• Vertical integration of κ∧κ over fibers of p: T(1) → T(1) yields the Weil-Petersson symplectic form on T(1), defined on H^{3/2} tangent vectors.
• For any cofinite Fuchsian group Γ, the same vertical integration on the Teichmüller curve F(Γ) recovers the Weil-Petersson symplectic form on T(Γ).
• The universal Teichmüller curve embeds into A∞(Δ) via a Bers-type embedding, with the image containing an open ball around the origin.