On the regularity of the structure of branched immersions

We show the existence of some special coordinate systems for expressing maps with branch points. These coordinates allow obtaining an explicit representation formula of branch immersions and understanding the regularity of some fundamental elements in the theory of R. D. Gulliver, R. Osserman and H. L. Royden. For conformal maps, we prove that these coordinates exist with certain regularity if and only if the mean curvature vector extends and is in a particular H\"older space. In the general case, we characterize this type of coordinate system. Also, using these coordinates, we study the regularity and behavior of curvatures near branch points.