On the stability of soliton and hairy black hole solutions of $\mathfrak{s}\mathfrak{u}(N)$ Einstein-Yang-Mills theory with a negative cosmological constant

We investigate the stability of spherically symmetric, purely magnetic, soliton and black hole solutions of four-dimensional $\mathfrak{s}\mathfrak{u}(N)$ Einstein-Yang-Mills theory with a negative cosmological constant $\mathrm{\Lambda }$. These solutions are described by $N-1$ magnetic gauge field functions $\omega _{j}$. We consider linear, spherically symmetric, perturbations of these solutions. The perturbations decouple into two sectors, known as the sphaleronic and gravitational sectors. For any $N$, there are no instabilities in the sphaleronic sector if all the magnetic gauge field functions $\omega _{j}$havenozeros,andsatisfyasetof$N-1$ inequalities. In the gravitational sector, we are able to prove that there are solutions which have no instabilities in a neighbourhood of stable embedded $\mathfrak{s}\mathfrak{u}(2)$ solutions, provided the magnitude of the cosmological constant $\left| \Lambda \right| $ is sufficiently large.