Morse Index of Willmore spheres in $S^3$

TL;DR

This paper establishes an upper bound for the Morse index of Willmore spheres in $S^3$, linking it to the quantized Willmore energy and the Jacobi operator of corresponding minimal surfaces.

Contribution

It provides a novel upper bound for the Morse index of Willmore spheres in $S^3$ based on their energy quantization and minimal surface correspondence.

Findings

Morse index is bounded above by the integer $m$ from energy quantization.

Explicit relation between second variation of Willmore energy and Jacobi operator.

Utilizes Bryant's classification of minimal surfaces via stereographic projection.

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 $\mathscr{W}(\Sigma)=4\pi m$. 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.