Extension of the Weil-Petersson connection

Abstract

Convexity properties of Weil-Petersson (WP) geodesics on the Teichmüller 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