Quasi-projections in Teichmüller space

Many parallels have been drawn between the geometric properties of the Teichmüller space of a Riemann surface, and those of complete, negatively curved spaces (see for example [B, K2, W]). This paper investigates one such parallel – the contracting properties of certain projections to geodesics. We will use the Teichmüller metric throughout the paper (the other famous metric on Teichmüller space, the Weil-Petersson metric, is negatively curved although it is not complete. The Teichmüller metric is complete, but not negatively curved – see [Ma].) Every closed subset C of a complete metric spaceX determines a “closest-points” projection, defined as a map πC : X → P(C), where P(C) is the set of closed subsets of C. Namely, πC(x) = {y ∈ C : d(x, y) = d(x,C)} where d(x,C) = infy∈C d(x, y). Suppose now that X is a simply-connected Riemannian manifold with non-positive sectional curvatures, and that C is a geodesic segment, ray or line. In this case πC(x) always contains a single point. If the sectional curvatures are bounded above by a negative constant, then |dπC | → 0 as d(x,C) → ∞. In other words, π is contracting at large distances. A standard consequence of this is that “quasi-geodesics” are always in bounded neighborhoods of actual geodesics in such a space (see section 3).