2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/138118Convexity properties of Weil-Petersson 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.Differential Geometry32G15 (Primary) 20H10, 30F60 (Secondary)text