Quasiconformal maps of bordered Riemann surfaces with L 2 Beltrami differentials

Let Σ be a Riemann surface of genus g bordered by n curves homeomorphic to the circle S 1 . Consider quasiconformal maps f : Σ→Σ 1 such that the restriction to each boundary curve is a Weil-Petersson class quasisymmetry. We show that any such f is homotopic to a quasiconformal map whose Beltrami differential is L 2 with respect to the hyperbolic metric on Σ. The homotopy H ( t , •): Σ → Σ 1 is independent of t on the boundary curves; that is, H ( t , p ) = f ( p ) for all p ∈ ∂Σ.