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Motivated by the novel asymptotically global AdS$_4$ solutions with deforminghorizon in [JHEP {\bf 1802}, 060 (2018)], we analyze the boundary metric withodd multipolar differential rotation and numerically construct a family ofdeforming solutions with tripolar differential rotation boundary, including twoclasses of solutions: solitons and black holes. We find that the maximal valuesof the rotation parameter $\varepsilon$, below which the stable large blackhole 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 timelikefor some regions of $\varepsilon<2$, solitons and black holes with tripolardifferential rotation could be unstable and develop hair due to superradiance.As the temperature $T$ drops toward $T_{schw}$, we find that though thereexists the spacelike Killing vector $\partial_{t}$ for some regions of$\varepsilon>2$, solitons and black holes still exist and do not develop hairdue to superradiance. Moreover, for $T\leqslant T_{schw}$, the curves ofentropy 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 blackholes by using an isometric embedding in the hyperbolic three-dimensionalspace. Furthermore, we also study the quasinormal modes of the solitons andblack holes, which have analogous behaviours to that of dipolar rotation andquadrupolar rotation.

Deforming black holes with odd multipolar differential rotation boundary