# Index and nullity of proper biharmonic maps in spheres

## TL;DR

This paper investigates the second variation of the bienergy functional for proper biharmonic maps in spheres, computing their index and nullity, and analyzing stability for certain non-compact domain maps.

## Contribution

It provides explicit calculations of index and nullity for known proper biharmonic maps and introduces a notion of equivariant variations for stability analysis.

## Key findings

- Explicit index and nullity values for specific proper biharmonic maps.
- Proper biharmonic maps from  to  are strictly stable under compactly supported variations.
- A new approach to stability analysis using equivariant variations.

## Abstract

In recent years, the study of the bienergy functional has attracted the attention of a large community of researchers, but there are not many examples where the second variation of this functional has been thoroughly studied. We shall focus on this problem and, in particular, we shall compute the exact index and nullity of some known examples of proper biharmonic maps. Moreover, we shall analyse a case where the domain is not compact. More precisely, we shall prove that a large family of proper biharmonic maps $\varphi:\mathbb{R} \to \mathbb{S}^2$ is strictly stable with respect to compactly supported variations. In general, the computations involved in this type of problems are very long. For this reason, we shall also define and apply to specific examples a suitable notion of index and nullity with respect to equivariant variations.