On willmore surfaces in Sn of flat normal bundle

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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 Sn must be located in some S3 ⊂ Sn, from which we characterize the Clifford torus as the only non-equatorial homogeneous minimal surface in Sn with flat normal bundle, which improves a result of K. Yang. Then we derive that every Willmore two sphere with flat normal bundle in Sn is conformal to a minimal surface with embedded planer ends in R3. 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 S3. In the end, we show that a Willmore surface with flat normal bundle must locate in some S6.

Original languageEnglish
Pages (from-to)3245-3255
Number of pages11
JournalProceedings of the American Mathematical Society
Issue number9
StatePublished - 2013
Externally publishedYes


  • Clifford torus
  • Flat normal bundle
  • S-Willmore surfaces
  • Willmore sphere
  • Willmore surfaces


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