Taiwanese Journal of Mathematics

WILLMORE SURFACES IN THE UNIT N-SPHERE

Yu-Chung Chang and Yi-Jung Hsu

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Abstract

Let $M^2$ be a compact Willmore surface in the $n$-dimensional unit sphere. Denote by $\phi_{ij}^{\alpha}$ the tracefree part of the second fundamental form $h_{ij}^{\alpha}$ of $M^2,$ and by $\mathbb{H}$ the mean curvature vector of $M^2.$ Let $\Phi$ be the square of the length of $\phi_{ij}^{\alpha}$ and $H=|\mathbb{H}|$. We prove that if $0\leq\Phi\leq C(1+\frac{H^2}{8})$, where $C=2$ when $n=3$ and $C=\frac{4}{3}$ when $n \geq 4$, then either $\Phi=0$ and $M^2$ is totally umbilic or $\Phi=C(1+\frac{H^2}{8})$. In the latter case, either $n=3$ and $M^2$ is the Clifford torus or $n=4$ and $M^2$ is the Veronese surface.

Article information

Source
Taiwanese J. Math., Volume 8, Number 3 (2004), 467-476.

Dates
First available in Project Euclid: 18 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500407666

Digital Object Identifier
doi:10.11650/twjm/1500407666

Mathematical Reviews number (MathSciNet)
MR2163319

Zentralblatt MATH identifier
1075.53052

Subjects
Primary: 53C42: Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42] 53A10: Minimal surfaces, surfaces with prescribed mean curvature [See also 49Q05, 49Q10, 53C42]

Keywords
Willmore surface Willmore functional totally umbilic sphere

Citation

Chang, Yu-Chung; Hsu, Yi-Jung. WILLMORE SURFACES IN THE UNIT N-SPHERE. Taiwanese J. Math. 8 (2004), no. 3, 467--476. doi:10.11650/twjm/1500407666. https://projecteuclid.org/euclid.twjm/1500407666


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References

  • R. Bryant, A duality theorem for Willmore surfaces, J. Differential Geom. 20 (1984), 23-53.
  • B. Y. Chen, Some conformal invariants of submanifolds and their applications, Boll. Un. Mat. Ital. 10 (1974), 380-385.
  • S. S. Chern, M. do Carmo, and S. Kobayashi, Minimal submanifold of a sphere with second fundamental form of constant length, Functional Analysis and Related Fields,Springer-Verlag, Berlin, 1970, 59-75.
  • M. Kozlowski, and U. Simon, Minimal immersions of 2-manifolds into spheres, Math. Z. 186 (1984), 377-382.
  • H. Li, Willmore hypersurfaces in a sphere, Asian J. Math. 5 (2001), 365-378.
  • H. Li, Willmore surfaces $S^n$, Ann. Global Anal. Geom. 21(2002), 203-213.
  • S. Montiel, and A. Ros, Minimal immersions of surfaces by the first eigenfunctions and conformal area, Invent. Math. 83 (1985), 153-166.
  • J. L. Weiner, On a problem of Chen, Willmore et al., Indiana Univ. Math. J. 27 (1978), 19-35.