On the Morse index of Willmore spheres in $S^3$

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, which is a consequence of the classification of Willmore spheres in $S^3$ by Robert Bryant, shows that there exists an integer $m$ such that $\mathscr{W} (\Sigma) = 4 \pi m$. We show that the Morse index $\operatorname{Ind}_\mathscr{W} (\Sigma)$ of the Willmore sphere $\Sigma$ satisfies the inequality $\operatorname{Ind}_\mathscr{W} (\Sigma) \leq m$.

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Received 14 May 2016

Accepted 27 February 2018

Published 2 December 2020