  Motivated by the novel asymptotically global AdS$_4$ solutions with deforming
horizon in [JHEP {\bf 1802}, 060 (2018)], we analyze the boundary metric with
odd multipolar differential rotation and numerically construct a family of
deforming solutions with tripolar differential rotation boundary, including two
classes of solutions: solitons and black holes. We find that the maximal values
of the rotation parameter $\varepsilon$, below which the stable large black
hole solutions could exist, are not a constant for $T>
T_{schw}=\sqrt{3}/2\pi\simeq0.2757$. When temperature is much higher than $
T_{schw}$, even though the norm of Killing vector $\partial_{t}$ keeps timelike
for some regions of $\varepsilon<2$, solitons and black holes with tripolar
differential rotation could be unstable and develop hair due to superradiance.
As the temperature $T$ drops toward $T_{schw}$, we find that though there
exists the spacelike Killing vector $\partial_{t}$ for some regions of
$\varepsilon>2$, solitons and black holes still exist and do not develop hair
due to superradiance. Moreover, for $T\leqslant T_{schw}$, the curves of
entropy firstly combine into one curve and then separate into two curves again,
in the case of each curve there are two solutions at a fixed value of
$\varepsilon$. In addition, we study the deformations of horizon for black
holes by using an isometric embedding in the hyperbolic three-dimensional
space. Furthermore, we also study the quasinormal modes of the solitons and
black holes, which have analogous behaviours to that of dipolar rotation and
quadrupolar rotation.
