Willmore surfaces in the unit N-sphere

Yu Chung Chang*, Yi-Jung Hsu

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

3 Scopus citations


Let M2 be a compact Willmore surface in the n-dimensional unit sphere. Denote by φij α the tracefree part of the second fundamental form φij α of M2, and by ℍ the mean curvature vector of M2. Let Φ be the square of the length of φij α and H = |ℍ|. We prove that if 0 ≤ Φ ≤ C(1 + H2/8), where (C = 2 when n = 3 and C = 4/3 when n ≥ 4, then either Φ = 0 and M2 is totally umbilic or Φ = (C(1 + H2/8). In the latter case, either n = 3 and M2 is the Clifford torus or n = 4 and M2 is the Veronese surface.

Original languageEnglish
Pages (from-to)467-476
Number of pages10
JournalTaiwanese Journal of Mathematics
Issue number3
StatePublished - 1 Jan 2004


  • Sphere
  • Totally umbilic
  • Willmore functional
  • Willmore surface

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