A convergent Weil-Petersson metric on the Teichm\"uller space of bordered Riemann surfaces

Abstract

Let $\Sigma$ be a Riemann surface of genus $g$ bordered by $n$ curves homeomorphic to the circle $\mathbb{S}^1$, and assume that $2g+2-n>0$. For such bordered Riemann surfaces, the authors have previously defined a Teichm\"uller space which is a Hilbert manifold and which is holomorphically included in the standard Teichm\"uller space. Based on this, we present alternate models of the aforementioned Teichm\"uller space and show in particular that it is locally modelled on a Hilbert space of $L^2$ Beltrami differentials, which are holomorphic up to a power of the hyperbolic metric, and has a convergent Weil-Petersson metric.