In this article we give a criterion for the existence of a metric of curvature 1 on a 2-sphere with n conical singularities of prescribed angles 2πϑ 1 , . . . , 2πϑ n and non-coaxial holonomy. Such a necessary and sufficient condition is expressed in terms of linear inequalities in ϑ 1 , . . . , ϑ n .

Abstract. Let T be Teichmüller space of a closed surface of genus at least 2. For any point c ∈ T , we describe an action of the circle on T × T , which limits to the earthquake flow when one of the parameters goes to a measured lamination in the Thurston boundary of T . This circle action shares some of the main properties of the earthquake flow, for instance it satisfies an extension of Thurston's Earthquake Theorem and it has a complex extension which is analogous and limits to complex earthquakes. Moreover, a related circle action on T × T extends to the product of two copies of the universal Teichmüller space.

Abstract. The landslide flow, introduced in [5], is a smoother analog of the earthquake flow on Teichmüller space which shares some of its key properties. We show here that further properties of earthquakes apply to landslides. The landslide flow is the Hamiltonian flow of a convex function. The smooth grafting map sgr taking values in Teichmüller space, which is to landslides as grafting is to earthquakes, is proper and surjective with respect to either of its variables. The smooth grafting map SGr taking values in the space of complex projective structures is symplectic (up to a multiplicative constant). The composition of two landslides has a fixed point on Teichmüller space. As a consequence we obtain new results on constant Gauss curvature surfaces in 3-dimensional hyperbolic or AdS manifolds. We also show that the landslide flow has a satisfactory extension to the boundary of Teichmüller space.

Given a hyperbolic surface with geodesic boundary S, the lengths of a maximal system of disjoint simple geodesic arcs on S that start and end at ∂S perpendicularly are coordinates on the Teichmüller space T (S). We express the Weil-Petersson Poisson structure of T (S) in this system of coordinates and we prove that it limits pointwise to the piecewise-linear Poisson structure defined by Kontsevich on the arc complex of S. At the same time, we obtain a formula for the first-order variation of the distance between two closed geodesics under Fenchel-Nielsen deformation.

We begin by describing the duality between arc systems and ribbon graphs embedded in a punctured surface and explaining how to cellularize the moduli space of curves in two different ways: using Jenkins-Strebel differentials and using hyperbolic geometry. We also briefly discuss how these two methods are related. Next, we recall the definition of Witten cycles and we illustrate their connection with tautological classes and Weil-Petersson geometry. Finally, we exhibit a simple argument to prove that Witten classes are stable.2000 Mathematics Subject Classification: 32G15, 30F30, 30F45.