On Willmore surfaces in 𝑆ⁿ of flat normal bundle

Wang, Peng

doi:10.1090/S0002-9939-2013-11683-7

On Willmore surfaces in $S^n$ of flat normal bundle
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Proc. Amer. Math. Soc. 141 (2013), 3245-3255 Request permission

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 be located in some $S^3\subset S^n$, from which we characterize the Clifford torus as the only non-equatorial homogeneous minimal surface in $S^n$ with flat normal bundle, which improves a result of K. Yang. Then we derive that every Willmore two sphere with flat normal bundle in $S^n$ 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 a flat normal bundle if and only if they are located in some $S^3$. In the end, we show that a Willmore surface with flat normal bundle must locate in some $S^6$.

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