We obtain an upper bound for the Morse index of Willmore spheres $\Sigma\subset S^3$ coming from an immersion of $S^2$. The quantization of Willmore energy shows that there exists an integer $m$ such that $\mathscr{W}(\Sigma)=4\pi m$. Then we show that $\mathrm{Ind}_{\mathscr{W}}(\Sigma)\leq m$. The proof relies on an explicit computation relating the second derivative of $\mathscr{W}$ for $\Sigma$ with the Jacobi operator of the minimal surface in $\mathbb{R}^3$ it is the image of by stereographic projection thanks of the fundamental classification of Robert Bryant.

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关于 $S^3$ 中 Willmore 球体的莫尔斯指数

我们从 $S^2$ 的浸入中获得了 Willmore 球体 $\Sigma\subset S^3$ 的莫尔斯指数的上限。Willmore 能量的量化表明存在一个整数 $m$，使得 $\mathscr{W}(\Sigma)=4\pi m$。然后我们证明 $\mathrm{Ind}_{\mathscr{W}}(\Sigma)\leq m$。该证明依赖于显式计算，将 $\Sigma$ 的 $\mathscr{W}$ 的二阶导数与 $\mathbb{R}^3$ 中的最小表面的 Jacobi 算子相关联，它是立体投影的图像感谢罗伯特·布莱恩特的基本分类。