Remarks on the boundary curve of a constant mean curvature topological disc

TL;DR

This paper explores the geometric properties of constant mean curvature topological discs with analytic boundaries, deriving formulas and conditions related to their boundary curvature and umbilicity.

Contribution

It introduces new formulas linking mean curvature to boundary normal curvature and provides conditions for total umbilicity based on boundary properties.

Findings

Derived a formula for mean curvature as a weighted average of boundary normal curvature

Established a condition for the surface to be totally umbilic based on boundary curvature

Analyzed consequences of the holomorphic quadratic Hopf differential on such surfaces

Abstract

We discuss some consequences of the existence of the holomorphic quadratic Hopf differential on a conformally immersed constant mean curvature topological disc with analytic boundary. In particular, we derive a formula for the mean curvature as a weighted average of the normal curvature of the boundary curve, and a condition for the surface to be totally umbilic in terms of the normal curvature.