# On the Index of Willmore spheres

## TL;DR

This paper investigates the Morse Index of Willmore spheres in Euclidean space, establishing bounds and explicit calculations, and reveals a connection between the index and the geometry of associated minimal surfaces.

## Contribution

It provides explicit formulas for the Morse Index of Willmore spheres and links it to the properties of the underlying minimal surfaces, including the number of ends and normal span dimension.

## Key findings

- The Morse Index of a Willmore sphere is m - d, with m the number of ends and d the span dimension of normals.
- For four ends, the span dimension d equals three.
- The Morse Index relates strongly to logarithmic Jacobi fields on the minimal surface.

## Abstract

We consider unbranched Willmore surfaces in the Euclidean space that arise as inverted complete minimal surfaces with embedded planar ends. Several statements are proven about upper and lower bounds on the Morse Index - the number of linearly independent variational directions that locally decrease the Willmore energy. We in particular compute the Index of a Willmore sphere in the three-space. This Index is $m-d$, where $m$ is the number of ends of the corresponding complete minimal surface and $d$ is the dimension of the span of the normals at the $m$-fold point. The dimension $d$ is either two or three. For $m=4$ we prove that $d=3$. In general, we show that there is a strong connection of the Morse Index to the number of logarithmically growing Jacobi fields on the corresponding minimal surface.