[Paper Review] On Teichmuller Space of Surface with Boundary

Feng Luo|ArXiv.org|Jan 15, 2006

Geometric Analysis and Curvature Flows18 references3 citations

TL;DR

This paper introduces a new parameterization of the bordered Teichmüller space of compact hyperbolic surfaces with boundary by using E-invariants derived from right-angled hyperbolic hexagons in an ideal triangulation. The key result is that the Teichmüller space with fixed boundary lengths becomes an open convex polytope in the E-invariant coordinates, and the existence of a hyperbolic metric is guaranteed if the E-invariants satisfy certain positivity and triangle-like inequalities derived from linear programming duality.

ABSTRACT

We characterization hyperbolic metrics on compact surfaces with boundary using a variational principle. As a consequence, a new parametrization of the Teichmuller space of compact surface with boundary is produced. In the new parametrization, the Teichmuller space becomes an open convex polytope. It is conjectured that the Weil-Petersson symplectic form can be expressed explicitly in terms of the new coordinate.

Motivation & Objective

• To develop a new coordinate system for the Teichmüller space of compact hyperbolic surfaces with non-empty boundary and fixed boundary lengths.
• To characterize hyperbolic metrics via a variational principle based on the cosine law for right-angled hyperbolic hexagons.
• To show that the bordered Teichmüller space is diffeomorphic to an open convex polytope under the new E-invariant coordinates.
• To establish necessary and sufficient conditions for the existence of a hyperbolic metric with prescribed E-invariants, using duality in linear programming.

Proposed method

• Define E-invariants for each edge in an ideal triangulation as half the sum of the lengths of opposite x-arcs minus the length of the facing edge.
• Use the energy function derived from the dilogarithm and cosine laws for right-angled hyperbolic hexagons to parameterize the Teichmüller space.
• Apply a variational principle: hyperbolic metrics correspond to critical points of the energy function on a convex domain defined by the E-invariants.
• Establish a one-to-one correspondence between hyperbolic metrics and E-invariant functions satisfying positivity and triangle-type inequalities.
• Use linear programming duality and Farkas’ lemma to prove that a non-empty solution space exists if and only if the E-invariants satisfy positivity on all essential simple loops.
• Show that the space of admissible E-invariants forms an open convex polytope in R^m, where m is the number of edges in the triangulation.

Experimental results

Research questions

• RQ1 Can the bordered Teichmüller space of a compact hyperbolic surface with boundary be parameterized explicitly using geometric invariants from ideal triangulations?
• RQ2 What conditions on the E-invariants ensure the existence of a hyperbolic metric with those invariants?
• RQ3 How does the variational principle based on the cosine law for right-angled hyperbolic hexagons characterize hyperbolic metrics on surfaces with boundary?
• RQ4 Is the bordered Teichmüller space with fixed boundary lengths diffeomorphic to an open convex polytope under the E-invariant parameterization?
• RQ5 Can the Weil-Petersson symplectic form be expressed explicitly in terms of the new E-invariant coordinates, as conjectured?

Key findings

• The bordered Teichmüller space T(S,l) with fixed boundary lengths is diffeomorphic to an open convex polytope in R^m, where m is the number of edges in an ideal triangulation.
• A hyperbolic metric exists on the surface if and only if the E-invariants satisfy the system of inequalities derived from the triangle-like relations in each hexagonal cell.
• The energy function V, constructed from the dilogarithm and cosine laws, achieves its maximum on the closure of the solution space, and the maximum lies in the interior if and only if a hyperbolic metric exists.
• The existence of a solution to the E-invariant system is equivalent to the positivity of the linear functional ∑ y_i z_i > 0 for all non-zero vectors y in the cone D of measured laminations.
• The E-invariant parameterization provides a new, explicit, and geometrically meaningful coordinate system for the bordered Teichmüller space, generalizing Leibon’s work on closed surfaces.
• The space of admissible E-invariants is characterized by linear inequalities derived from the geometry of right-angled hyperbolic hexagons and the structure of the triangulation.