Bending Fuchsian representations of fundamental groups of cusped surfaces in PU(2,1)

We describe a family of representations of $\pi_1(\Sigma)$ in PU(2,1), where $\Sigma$ is a hyperbolic Riemann surface with at least one deleted point. This family is obtained by a bending process associated to an ideal triangulation of $\Sigma$. We give an explicit description of this family by describing a coordinates system in the spirit of shear coordinates on the Teichm\"uller space. We identify within this family new examples of discrete, faithful and type-preserving representations of $\pi_1(\Sigma)$. In turn, we obtain a 1-parameter family of embeddings of the Teichm\"uller space of $\Sigma$ in the PU(2,1)-representation variety of $\pi_1(\Sigma)$. These results generalise to arbitrary $\Sigma$ the results obtained in a previous paper for the 1-punctured torus.