Understanding Weil-Petersson curvature

2008/09/22 by Scott A. Wolpert, Wolpert, Scott A.
Mathematics · #30F60 (Secondary) #32G15 (Primary) 20H10 #Analytic and geometric function theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Geometric and Algebraic Topology

paper · pdf · doi:10.48550/arxiv.0809.3699

openalex publication_date 2008/09/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A brief history of the investigation of the Weil-Petersson curvature and a summary of Teichmüller theory are provided. A report is presented on the program to describe an intrinsic geometry with the Weil-Petersson metric and geodesic-length functions. Formulas for the metric, covariant derivative and formulas for the curvature tensor are presented. A discussion of methods is included. Recent and new applications are sketched, including results from the work of Liu-Sun-Yau, an examination of the Yamada model metric and a description of Jacobi fields along geodesics to the boundary.

Citations

• The Weil-Petersson metric geometry

Related

• The Weil-Petersson metric geometry
• Average curvatures of Weil-Petersson geodesics in Teichmuller space
• Local and Global Aspects of Weil-Petersson Geometry
• Families of Riemann surfaces and Weil-Petersson geometry
• Geodesic-length functions and the Weil-Petersson curvature tensor