Gauss maps of M\"obius surfaces in the $n$-dimensional sphere

TL;DR

This paper explores Gauss maps for M"obius surfaces in the n-sphere, linking their harmonicity to Willmore surface conditions and clarifying previous research, with applications to the Bj"orling problem.

Contribution

It introduces the concept of Lorentzian 2-plane lifts for M"obius surfaces and establishes their conformal harmonicity as equivalent to the Willmore condition, extending prior work.

Findings

Lorentzian 2-plane lift's harmonicity characterizes Willmore surfaces.

Clarifies previous results by Hélien, Xia-Y Shen, and Ma.

Enables treatment of the Bj"orling problem with umbilics.

Abstract

In this note we discuss Gauss maps for M\"obius surfaces in the $n$-sphere, and their applications in the study of Willmore surfaces. One such ``Gauss map'', naturally associated to a Willmore surface that has a dual Willmore surface, is the Lorentzian $2$-plane bundle given by a lift of the suface and its dual. More generally, we define the concept of a Lorentzian $2$-plane lift for an arbitrary M\"obius surface, and show that the conformal harmonicity of this lift is equivalent to the Willmore condition for the surface. This clarifies some previous work of F. H\'elein, Q. Xia-Y Shen, X. Ma and others, and, for instance, allows for the treatment of the Bj\"orling problem for Willmore surfaces in the presence of umbilics.