Circle packings as differentiable manifolds

Circle packings are configurations of circles satisfying specified patterns of tangency and have emerged as the foundation for a fairly comprehensive theory of discrete analytic functions. Though many classical results found their counterpart in circle packing, other concepts have not yet been transferred, particularly those which require a linear structure. This paper puts circle packings in a framework of smooth manifolds, providing access to linear structures in their tangent spaces. Since we are especially interested in applications to boundary value problems (of Beurling and Riemann–Hilbert type), we do not only investigate the manifolds of circle packings and packing labels, but also manifolds formed by the centers of the boundary circles. The approach is elementary and rests on a detailed analysis of the contact equations which govern the tangency relation between neighboring circles.