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

Quasiconformal maps of bordered Riemann surfaces with L 2 Beltrami differentials