Extension of the Weil-Petersson connection

TL;DR

This paper investigates the convexity properties of Weil-Petersson geodesics on Teichmüller space, introduces a normal form for the Levi-Civita connection, and applies it to analyze geodesic approximation and convexity of horocycle distances.

Contribution

It presents a new normal form for the Weil-Petersson Levi-Civita connection and applies it to study geodesic convexity and boundary approximations.

Findings

Convexity of the distance between horocycles along Weil-Petersson geodesics.

A normal form for the Weil-Petersson Levi-Civita connection for pinched hyperbolic metrics.

Approximation of geodesics in boundary spaces.

Abstract

Convexity properties of Weil-Petersson geodesics on the Teichm\"{u}ller space of punctured Riemann surfaces are investigated. A normal form is presented for the Weil-Petersson Levi-Civita connection for pinched hyperbolic metrics. The normal form is used to establish approximation of geodesics in boundary spaces. Considerations are combined to establish convexity along Weil-Petersson geodesics of the functions the distance between horocycles for a hyperbolic metric.