  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.
