## Taiwanese Journal of Mathematics

### WILLMORE SURFACES IN THE UNIT N-SPHERE

#### 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

https://projecteuclid.org/euclid.twjm/1500407666

Digital Object Identifier
doi:10.11650/twjm/1500407666

Mathematical Reviews number (MathSciNet)
MR2163319

Zentralblatt MATH identifier
1075.53052

#### 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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