On Willmore surfaces in S^n of flat normal bundle

TL;DR

This paper investigates Willmore surfaces with flat normal bundle in spheres, showing their localization in lower-dimensional spheres and characterizing specific types like Clifford tori and minimal surfaces with planar ends.

Contribution

It characterizes Willmore surfaces with flat normal bundle in spheres, revealing their confinement to lower-dimensional spheres and identifying unique examples such as Clifford tori.

Findings

S-Willmore surfaces with flat normal bundle lie in some S^3

Clifford torus is the only non-equatorial homogeneous minimal surface with flat normal bundle in S^n

Willmore two-spheres with flat normal bundle are conformal to minimal surfaces with planar ends in R^3

Abstract

We discuss several kinds of Willmore surfaces of flat normal bundle in this paper. First we show that every S-Willmore surface with flat normal bundle in $S^n$ must locate in some $S^3\subset S^n$, from which we characterize Clifford torus as the only non-equatorial homogeneous minimal surface in $S^n$ with flat normal bundle, which improve a result of K. Yang. Then we derived that every Willmore two sphere with flat normal bundle in $S^n$ is conformal to a minimal surface with embedded planer ends in $\mathbb{R}^3$. We also point out that for a class of Willmore tori, they have flat normal bundle if and only if they locate in some $S^3$. In the end, we show that a Willmore surface with flat normal bundle must locate in some $S^6$