Understanding Weil-Petersson curvature

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.

References in corpus (6)

• Geometry of the Weil-Petersson completion of Teichmüller space
• The Weil-Petersson metric geometry
• Positivity of relative canonical bundles for families of canonically polarized manifolds
• Behavior of geodesic-length functions on Teichmueller space
• Extension of the Weil-Petersson connection
• Average curvatures of Weil-Petersson geodesics in Teichmuller space

Cited by in corpus (5)

• Strict and non strict positivity of direct image bundles
• Recent Development on the Geometry of the Teichmuller and Moduli Spaces of Riemann Surfaces and Polarized Calabi-Yau Manifolds
• Asymptotics of a class of Weil-Petersson geodesics and divergence of Weil-Petersson geodesics
• Topological dynamics of the Weil-Petersson geodesic flow
• On the Weil-Petersson curvature of the moduli space of Riemann surfaces of large genus