## Taiwanese Journal of Mathematics

### WILLMORE SURFACES AND F-WILLMORE SURFACES IN SPACE FORMS

Yu-Chung Chang

#### Abstract

Let $M^{2}$ be a compact $F$-Willmore surface in the $n$-dimensional space form $\mathbb{N}^{n}(c)$ of constant curvature $c$. Denote by $\phi_{ij}^{\alpha}$ the trace free part of the second fundamental form $h=(h_{ij}^{\alpha})$, and by $\mathbf{H}$ the mean curvature vector of $M^{2}$. Let $\Phi$ be the square of the length of $\phi_{ij}^{\alpha}$ and $H = |\mathbf{H}|$. If $F^{\prime}(\Phi)\geq 0$, then $\int_{M} \Big\{F^{\prime\prime}(\Phi)\Big[\frac{1}{2}|\nabla\Phi|^{2} - \sum_{\alpha,i,j} \phi_{ij}^{\alpha}\Phi_{j}H_{i}^{\alpha}\Big]+F(\Phi)H^{2} + F^{\prime}(\Phi) (2c-K(n)\Phi)\Phi\Big\} dv \leq 0$. The constant function $K(n) = 1$ when $n=3$ and $K(n) = \frac{3}{2}$ when $n\geq 4$. Similarly, $\int_{M} \Big\{F^{\prime\prime}(\Phi) \Big[\frac{1}{2}|\nabla\Phi|^{2} - \sum_{\alpha,i,j} \phi_{ij}^{\alpha} \Phi_{j}H_{i}^{\alpha}\Big] + F(\Phi)H^{2} + F^{\prime}(\Phi) (2c-K(n)\Phi)\Phi\Big\} dv \geq 0$, if $F^{\prime}(\Phi) \leq 0$. We also prove the following: If $M^{2}$ is a compact Willmore surface in the $n$-dimensional space form $\mathbb{N}^{n}(c)$. Then $\int_{M}\Phi (C(n)(c+\frac{H^{2}}{2})-\Phi)\leq 0$, where $C(n)=2$ when $n=3$ and $C(n)=\frac{4}{3}$ when $n\geq 4$. If $0\leq\Phi\leq C(n)(c+\frac{H^{2}}{2})$, then either $\Phi=0$ and $M$ is totally umbilical sphere, or $\Phi=C(n)(c+\frac{H^{2}}{2})$. In the latter case, either $M$ is the Clifford torus in $S^{3}$ of $\mathbb{N}^{n}(c)$, or $M$ is the Veronese surface in $S^{4}$ of $\mathbb{N}^{n}(c)$.

#### Article information

Source
Taiwanese J. Math., Volume 17, Number 1 (2013), 109-131.

Dates
First available in Project Euclid: 10 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1499705878

Digital Object Identifier
doi:10.11650/tjm.17.2013.1840

Mathematical Reviews number (MathSciNet)
MR3028860

Zentralblatt MATH identifier
1267.53058

#### Citation

Chang, Yu-Chung. WILLMORE SURFACES AND F-WILLMORE SURFACES IN SPACE FORMS. Taiwanese J. Math. 17 (2013), no. 1, 109--131. doi:10.11650/tjm.17.2013.1840. https://projecteuclid.org/euclid.twjm/1499705878

#### References

• R. Bryant, A duality theorem for Willmore surfaces, J. Differential Geom., 20 (1984), 23-53.
• M. Cai, $L^{p}$ Willmore functionals, Proc. Am. Math. Soc., 127 (1999), 569-575; Differential Geom., 20 (1984), 23-53.
• S. S. Chern, M. P. do Carmo and S. Kobayashi, Minimal submanifold of a sphere with second fundamental form of constant length, Functional Analysis and Related Fields, Springer-Verlag, 1970, pp. 59-75.
• B. Y. Chen, Some conformal invariants of submanifolds and their applications, Boll. Un. Mat. Ital., 10 (1974), 380-385.
• Y. C. Chang and Y. J. Hsu, Willmore surfaces in the unit n-sphere, Taiwanese J. of Math., 8 (2004), 467-476.
• Y. C. Chang and Y. J. Hsu, A pinching theorem for conformal classes of Willmore surfaces in the unit 3-sphere, Bulletin of the Institute of Mathematics Academia Sinica New Series, 2006.
• 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 Journal of Math., 5 (2001), 365-378.
• H. Li, Willmore surfaces in $S^n$, Ann. Global Anal. Geom., 21 (2002), 203-213.
• J. Liu and H. Jian, F-Willmore submanifold in space forms, Front. Math. China, 6 (2011), 871-886.
• Z. Hu and H. Li, Willmore submanifolds in a Riemannian manifold, Proceedings of the workshop Contemporary Geometry and Related Topics, World Scientific, September 15, 2003, pp. 275-299.
• P. Li and S. T. Yau, A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces, Invent. Math., 69 (1982), 269-291.
• W. Santos, ubmanifolds with parallel mean curvature vector in spheres, Tohoku Math. J., 46 (1994), 403-415.
• J. Simons, Minamal varieties in riemannian manifolds, Ann. of Math., 88 (1968), 62-105.
• J. L. Weiner, On a problem of Chen, Willmore et al., Indiana Univ. Math. J., 27 (1978), 19-35.