  We consider solutions in Einstein-Maxwell theory with a negative cosmological
constant that asymptote to global $AdS_{4}$ with conformal boundary
$S^{2}\times\mathbb{R}_{t}$. At the sphere at infinity we turn on a
space-dependent electrostatic potential, which does not destroy the asymptotic
$AdS$ behaviour. For simplicity we focus on the case of a dipolar electrostatic
potential. We find two new geometries: (i) an $AdS$ soliton that includes the
full backreaction of the electric field on the $AdS$ geometry; (ii) a polarised
neutral black hole that is deformed by the electric field, accumulating
opposite charges in each hemisphere. For both geometries we study boundary data
such as the charge density and the stress tensor. For the black hole we also
study the horizon charge density and area, and further verify a Smarr formula.
Then we consider this system at finite temperature and compute the Gibbs free
energy for both $AdS$ soliton and black hole phases. The corresponding phase
diagram generalizes the Hawking-Page phase transition. The $AdS$ soliton
dominates the low temperature phase and the black hole the high temperature
phase, with a critical temperature that decreases as the external electric
field increases. Finally, we consider the simple case of a free charged scalar
field on $S^{2}\times\mathbb{R}_{t}$ with conformal coupling. For a field in
the $SU(N)$ adjoint representation we compare the phase diagram with the above
gravitational system.
