Geometry of the Weil-Petersson completion of Teichmüller space

Abstract

We present a view of the current understanding of the geometry of Weil-Petersson (WP) geodesics on the completion of the Teichmüller space. We sketch a collection of results by other authors and then proceed to develop the properties of the WP CAT(0) geometry. Our approach includes a simplified proof of the Masur-Wolf theorem, a classification of flats and of geodesic limits.

50 pages

References in corpus (1)

• Curvature and rank of Teichmüller space

Cited by in corpus (25)

• Asymptotic geometry of the mapping class group and Teichmueller space
• Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmuller geodesic flow
• Thurston's pullback map on the augmented Teichmüller space and applications
• The Weil-Petersson metric geometry
• A transfer principle: from periods to isoperiodic foliations
• Classifying spaces for proper actions of mapping class groups
• Growth of the Weil-Petersson inradius of moduli space
• Understanding Weil-Petersson curvature
• Rates of mixing for the Weil-Petersson geodesic flow I: no rapid mixing in non-exceptional moduli spaces
• Recurrent Weil-Petersson geodesic rays with non-uniquely ergodic ending laminations
• The Weil-Petersson gradient flow of renormalized volume and 3-dimensional convex cores
• Convexity of geodesic-length functions: a reprise
• Behavior of geodesic-length functions on Teichmueller space
• Recent Development on the Geometry of the Teichmuller and Moduli Spaces of Riemann Surfaces and Polarized Calabi-Yau Manifolds
• Extension of the Weil-Petersson connection
• Products of Farey graphs are totally geodesic in the pants graph
• Asymptotics of Weil-Petersson geodesics II: bounded geometry and unbounded entropy
• Augmented Teichmuller Spaces and Orbifolds
• From harmonic maps to the nonlinear supersymmetric sigma model of quantum field theory. At the interface of theoretical physics, Riemannian geometry and nonlinear analysis
• Cusps and the family hyperbolic metric
• Large flats in the pants graph
• Harmonic Maps and Teichmueller Theory
• Families of Line Bundles over Riemann Surfaces, their Sections, and their Degenerations -- A Constructive Approach using Automorphic Forms
• Asymptotics of the Gaussian Curvatures of the Canonical Metric on the Surface
• Riemann moduli spaces are quantum ergodic