On the regularity of the structure of branched immersions

TL;DR

This paper introduces special coordinate systems for branched immersions, enabling explicit formulas and regularity analysis, especially near branch points, with implications for conformal maps and curvature behavior.

Contribution

It establishes the existence and regularity conditions of coordinate systems for branched immersions, enhancing understanding of their structure and curvature near branch points.

Findings

Existence of special coordinate systems for branched immersions.

Explicit representation formulas for branch immersions.

Characterization of regularity and curvature behavior near branch points.

Abstract

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.