On Willmore surfaces in 𝑆ⁿ of flat normal bundle

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