On Willmore surfaces in \(S^n\) of flat normal bundle (Q2845472)

The topic of this article are Willmore surfaces in \(S^n\) with flat normal bundle. This study is motivated by similar investigations on Willmore surfaces that satisfy the stronger condition of isothermicity.NEWLINENEWLINEThe author proves that S-Willmore surfaces with a flat normal bundle are contained in \(S^3\). As a corollary, minimal surfaces with flat normal bundle in a space of constant curvature must be contained in a three-dimensional subspace. Improving a result of [\textit{K. Yang}, Proc. Am. Math. Soc. 94, 119--122 (1985; Zbl 0535.53049)], the author shows that every non-equatorial homogeneous minimal surface in \(S^n\) with flat normal connection is isometric to the Clifford torus in \(S^3\). He then proves that two-dimensional Willmore surfaces with flat normal bundle and Willmore surfaces with flat normal bundle obtained from the Hopf bundle are necessarily contained in \(S^3\). The article concludes with some open questions in similar spirit.