Some global properties of umbilic points of Willmore immersions in the $3$-sphere

Abstract

We study the umbilic points of Willmore surfaces in codimension 1 from the viewpoint of the conformal Gauss map. We first study the local behaviour of the conformal Gauss map near umbilic curves and prove that they are geodesics up to a conformal transformation if and only if the Willmore immersion is, up to a conformal transformation, the gluing of minimal surfaces in the 3-dimensional hyperbolic space. Then, we prove a Gauss--Bonnet formula for the conformal Gauss map of Willmore surfaces which turns out to be an asymptotic expansion involving the length of the umbilic curves in the spirit of renormalized volume expansions. We interpret this formula as a unified version for the different expressions of the value of the Willmore energy for conformally minimal surfaces in each space-form.