Complete embedded constant mean curvatures (CMC) surfaces in R^3 seem to be highly transcendental objects, and their moduli spaces are generally only understood in a few special cases. In this talk, we will reveal a surprising connection with complex projective structures and holomorphic solutions to Hill's equation U_zz + q(z)U = 0 , where q(z)is a polynomial on C (really, a holomorphic quadratic differential Q = q(z)dz^2, obtained by taking the Schwartzian of the developing map for the projective structure). This allows us to explicitly work out the moduli space of k-ended coplanar CMC surfaces of genus 0 in terms of k-point projective structures on C, that is, those projective structures whose associated curvature -1 metrics have exactly k completion points. The latter is shown to be biholomorphic to the affine space of (monic, normalized) holomorphic quadratic differentials on C with polynomial growth of degree k-2, that is, to C^{k-3}; the former is thus diffeomorphic to R^{2k-3} = H^3 times C^{k-3}. We'll also discuss some related potential applications, including an explicit description of minimal surfaces in S^3.