Morse Index of Willmore spheres in $S^3$

Abstract

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 $\mathcal{W}(\Sigma )=4\pi m$. Then we show that ${\mathrm{Ind}}_{\mathcal{W}}(\Sigma )\le m$. The proof relies on an explicit computation relating the second derivative of $\mathcal{W}$ 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.