Handbook of Teichmüller Theory, Volume II

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pp: 47–64

DOI: 10.4171/055-1/2

The Weil–Petersson metric geometry

Scott A. Wolpert[1]

(1) Department of Mathematics, University of Maryland, MD 20742, COLLEGE PARK, UNITED STATES

The aim is to present a summary introduction for the current understanding of a Weil–Petersson metric space. We present an exposition following the development of concepts rather than an exposition following the order of results discovered. Selected readings and general attributions are provided at the end of each section.

Teichmüller space and its augmentation are described in terms of the Fenchel–Nielsen coordinates. The Weil–Petersson metric space basic properties are described. Geodesic-length functions are introduced and formulas for their gradients and Hessians are presented. Beginning applications are considered. A description of the Weil–Petersson metric in Fenchel–Nielsen coordinates is presented. A model metric is discussed. Properties of Weil–Petersson metric and geodesics, as well as properties of geodesic-length functions are combined to describe the Alexandrov tangent cone at points of the augmentation. A comparison dictionary is presented between the geometry of the space of flat tori and Teichmüller space with the Weil–Petersson metric.

Keywords: Weil–Petersson metric, CAT(0) space, geodesic length function, Weil–Petersson convexity, Fenchel–Nielsen coordinates, Alexandrov tangent cone