Understanding the space of all 'soap bubbles' - that is, complete embedded constant mean curvatures (CMC) surfaces in $R^3$ - is a central problem in geometric analysis. These CMC surfaces are highly transcendental objects; the topology and smooth structure of their moduli spaces are understood only in some special cases. In this talk we will describe the formal 'Lagrangian embedding' of CMC moduli space into the 'space of asymptotes' and discuss where this is smooth, namely, at a surface with no nontrivial square-integrable Jacobi fields. This nondegeneracy condition has now been established for all coplanar CMC surfaces of genus zero; this allows them to serve as 'building blocks' for more complicated CMC surfaces. There is also a surprising connection with complex projective structures and holomorphic quadratic differentials on $C$ obtained by taking the Schwarzian of the developing map for the projective structure. This assigns each coplanar CMC surface a 'classifying' complex polynomial, and lets us explicitly work out the smooth topology of their moduli spaces.