Rigidity theorem forWillmore surfaces in a sphere
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Let $M^{2}$ be a compact Willmore surface in the $(2 + p)$-dimensional unit sphere $S^{2+p}$. Denote by $H$ and $S$ the mean curvature and the squared length of the second fundamental form of $M^{2}$, respectively. Set $\rho^{2} = S − 2H^{2}$. In this note, we proved that there exists a universal positive constant $C$, such that if $\parallel \rho^{2}\parallel_{2}$ < $C$, then $\rho^{2} = 0$ and $M^{2}$ is a totally umbilical sphere.
Volume 132, 2022
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