Morse Index of Willmore spheres in $S^3$

We obtain an upper bound for the Morse index of Willmore spheres $Σ\subset S^3$ coming from an immersion of $S^2$. The quantization of Willmore energy shows that there exists an integer $m$ such that $\mathscr{W}(Σ)=4πm$. Then we show that $\mathrm{Ind}_{\mathscr{W}}(Σ)\leq m$. The proof relies on an explicit computation relating the second derivative of $\mathscr{W}$ for $Σ$ 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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