The umbilic set of Willmore surfaces

TL;DR

This paper investigates the structure of umbilic points on Willmore surfaces in codimension one, showing they form either a real-analytic curve or isolated points, extending understanding beyond minimal surfaces.

Contribution

It proves that the umbilic set on Willmore surfaces in codimension one is either a real-analytic curve or isolated points, generalizing known results for minimal surfaces.

Findings

Umbilic set is a real-analytic 1D manifold or isolated points

Generalizes minimal surface umbilic point results

Provides local structure theorem for umbilic points

Abstract

It is well known that the umbilic points of minimal surfaces in spaces of constant sectional curvature consist only of isolated points unless the surface is totally umbilic on some connected component, as for example the Hopf form is holomorphic. In this note, we prove that on Willmore surfaces in codimension one the umbilic set is locally a one dimensional real-analytic manifold without boundary or an isolated point.