Families of Riemann Surfaces and Weil-Petersson Geometry

This book is the companion to the CBMS lectures of Scott Wolpert at Central Connecticut State University. The lectures span across areas of research progress on deformations of hyperbolic surfaces and the geometry of the Weil-Petersson metric. The book provides a generally self-contained course for graduate students and postgraduates. The exposition also offers an update for researchers; material not otherwise found in a single reference is included. A unified approach is provided for an array of results. The exposition covers Wolpert’s work on twists, geodesic-lengths and the Weil-Petersson symplectic structure; Wolpert’s expansions for the metric, its Levi-Civita connection and Riemann tensor. The exposition also covers Brock’s twisting limits, visual sphere result and pants graph quasi isometry, as well as the Brock-Masur-Minsky construction of ending laminations for Weil-Petersson geodesics. The rigidity results of Masur-Wolf and Daskalopoulos-Wentworth, following the approach of Yamada, are included. The book concludes with a generally self-contained treatment of the McShane-Mirzakhani length identity, Mirzakhani’s volume recursion, approach to Witten-Kontsevich theory by hyperbolic geometry, and prime simple geodesic theorem. Lectures begin with a summary of the geometry of hyperbolic surfaces and approaches to the deformation theory of hyperbolic surfaces. General expositions are included on the geometry and topology of the moduli space of Riemann surfaces, the $CAT(0)$ geometry of the augmented Teichmüller space, measured geodesic and ending laminations, the deformation theory of the prescribed curvature equation, and the Hermitian description of Riemann tensor. New material is included on estimating orbit sums as an approach for the potential theory of surfaces. A co-publication of the AMS and CBMS.

Table of Contents

Chapter 1. Preliminaries

Chapter 2. Teichm¨uller Space and Horizontal Strip Deformations

Chapter 3. Geodesic-Lengths, Twists and Symplectic Geometry

Chapter 4. Geometry of the Augmented Teichm¨uller Space, Part 1

Chapter 5. Geometry of the Augmented Teichm¨uller Space, Part 2

Chapter 6. Geometry of the Augmented Teichm¨uller Space, Part 3

Chapter 7. Deformations of hyperbolic metrics and the curvature tensor

Chapter 8. Collar expansions and exponential-distance sums

Chapter 9. Train tracks and the Mirzakhani volume recursion

Chapter 10. Mirzakhani prime simple geodesic theorem