  A quantization of a breathing motion of a rotating chiral soliton on $S^3$ is
performed in terms of a family of trial functions for a profile function of the
hegdehog ansatz. We determine eigenenergies of the quantized $S^3$ skyrmion by
solving the Schr\"odinger equation of the breathing mode for several lower spin
and isospin states varying the Skyrme term constants $e$. When $S^3$ radius is
smaller than $2/ef_\pi$, where $f_\pi$ is the pion decay constant, we always
obtain a conformal map solution as the lowest eigenenergy state. In the
conformal map case, allowed states have only symmetric or anti-symmetric wave
function under inversion of a dynamical variable describing the breathing mode.
As the $S^3$ radius increases the energy splitting between the symmetric and
anti-symmetric states rapidly decreases and two states become completely
degenerate state. When the $S^3$ radius larger than $3/ef_\pi$, for the small
Skyrme term constant $e$ the lowest eigenenergy states are obtained with the
profile function given by an arccosine form which is almost the same to those
of usual $R^3$ skyrmion. When the effects of the Skyrme term are weak, i.e.
large $e$, the lowest energy states are obtained by the profile function of
conformal map, which correspond to the \lc\lc frozen states" for the $R^3$
skyrmion as the limit of $S^3$ radius $ \to \infty$.
