Stable proper biharmonic maps in Euclidean spheres

TL;DR

This paper constructs explicit examples of stable proper weak biharmonic maps from high-dimensional unit balls to Euclidean spheres, establishing their stability through second variation analysis, and is the first of its kind for maps from compact domains.

Contribution

It provides the first explicit stable proper weak biharmonic maps from compact domains into Euclidean spheres, with a detailed stability analysis and second variation formula derivation.

Findings

First example of stable proper weak biharmonic maps from a compact domain.

Explicit family of stable proper biharmonic maps constructed.

Second variation formula for bienergy established.

Abstract

We construct an explicit family of stable proper weak biharmonic maps from the unit ball $B^m$, $m\geq 5$, to Euclidean spheres. To the best of the authors knowledge this is the first example of a stable proper weak biharmonic map from at compact domain. To achieve our result we first establish the second variation formula of the bienergy for maps from the unit ball into a Euclidean sphere. Employing this result, we examine the stability of the proper weak biharmonic maps $q:B^m\to\mathbb{S}^{m^{\ell}}$, $m,\ell\in\mathbb{N}$ with $\ell\leq m$, which we recently constructed in \cite{BS25} and thus deduce the existence of an explicit family of stable proper biharmonic maps to Euclidean spheres.