Let M be a complete embedded CMC surface of finite topology (finitegenus g and number of ends k). Main Theorem: Assume M is contained in a half-space, i.e. coplanar,and that M has genus g = 0 (a condition we can likely drop). Then M is nondegenerate, i.e. the only square integrable Jacobi function on Mis identically zero.Corollary 1: The classifying map from the submoduli space \M'_{0,k} of such coplanar surfaces to the (2k-3)-dimensional manifold of sphericalk-point metrics (a classifying space locally modelled on the k-fold product of 2-spheres, modulo the diagonal action of the rotation groupSO(3)) is a real analytic diffeomorphism. Corollary 2: The asymptotes map from the moduli space \M_{0,k} of such CMC surfaces is an immersion in a neighborhood of the submoduli space\M'_{0,k}.The proof makes use of a "nonabelian" version of the Dirichlet-Neumann transform for harmonic functions (a.k.a. Hodge star on for harmonic 1-forms) on a surface and its linearization for Jacobi fields.If time permits, we will also discuss the synthetic geometry of the classifying space of spherical k-point metrics (and thus of the CMCmoduli space \M'_{0,k}) from several perspectives: a classical appproach using the Schwartzian derivative of developing map and corresponding holomorphic quadratic differentials; a tropical approach which makes use of the associahedron corresponding to various triangulations of a spherical k-gon; and a Techmuller-Thurston-like approach using the cross-ratio to compute explicitly. All three methods imply:Theorem 3. The classifying space of spherical k-point metrics is a(2k-3)-ball.