$SO(10)$ thick branes and perturbative stability

Three self-gravitating $SO(10)$ domain walls in five dimensions are obtained and their properties are analyzed. These non-abelian domain walls interpolate between AdS${}_5$ spacetimes with different embedding of $SU(5)$ in $SO(10)$ and they can be distinguished, among other features, by the unbroken group on each wall, being either $SO(10)$, $SO(6)\otimes SU(2)\otimes U(1)/Z_2$ or $SU(4)\otimes SO(2)\otimes U(1)/Z_4$. We show that, unlike Minkowskian versions, the curved scenarios are perturbatively stable due to the gravitational capture of scalar fluctuations associated to the residual orthogonal subgroup in the core of the walls. These stabilizer modes are additional to the four-dimensional Nambu-Goldstone states found in two of the three gravitational sceanarios.