Given a surface embedded in 3D Euclidean space, S → ℝ3, intrinsically, the surface has four layers of geometric information: topology, conformal structure, Riemannian metric, and embedding, corresponding to four geometries: topology, conformal geometry, Riemannian geometry, and diŠerential geometry for surfaces in ℝ3. In order to represent the topology, genus and the number of boundaries are required; for conformal geometry, 6g - 6 (or two) parameters are needed to describe the conformal structure of a genus g > 1 surface (or a torus). All genus zero closed surfaces have the same conformal structure. Given the conformal structure of S, a canonical conformal domain of S can be uniquely determined, denoted as DS; for Riemannian geometry, a function defined on the conformal domain λS Ds → ℝ is needed to specify the Riemannian metric; and finally, by adding a mean curvature function Hs:DS → ℝ, the embedding of S in 3 can be determined unique up to a rigid motion. erefore, in order to represent a 3D surface, one needs a finite number of parameters to determine a canonical domain DS, then two functions?, H defined on the domain. We denote this representation as (Ds,λs, Hs), and call it conformal representation.
