Stable proper biharmonic maps in Euclidean spheres

We construct an explicit family of stable proper weak biharmonic maps from the unit ball $B^m$, $m\ge 5$, 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 $q:B^m\to {\mathbb{S}}^{m^{\mathrm{\ell }}}$, $m,\mathrm{\ell }\in \mathbb{N}$ with $\mathrm{\ell }\le m$, which we recently constructed in \cite{BS25} and thus deduce the existence of an explicit family of stable proper biharmonic maps to Euclidean spheres.