# On the Morse Index of Branched Willmore Spheres in $3$-Space

## TL;DR

This paper introduces a method to compute the Morse index of branched Willmore spheres in 3-space, linking it to the index of a matrix related to the dual minimal surface's ends, and provides bounds for the index of certain Willmore spheres.

## Contribution

A general method to compute the Morse index of branched Willmore spheres and a relation to the dual minimal surface's ends are established.

## Key findings

- Morse index equals the index of a matrix related to the dual minimal surface.
- For Willmore spheres with energy 4πn, the Morse index is at most n-1.
- The method simplifies the computation of Morse indices for these spheres.

## Abstract

We develop a general method to compute the Morse index of branched Willmore spheres and show that the Morse index is equal to the index of certain matrix whose dimension is equal to the number of ends of the dual minimal surface. As a corollary, we find that for all immersed Willmore spheres $\vec{\Phi}:S^2\rightarrow \mathbb{R}^3$ such that $W(\vec{\Phi})=4\pi n$, we have $\mathrm{Ind}_{W}(\vec{\Phi})\leq n-1$