  The axial-vector form factors and axial-vector constants of the baryon
decuplet are investigated within a pion mean-field approach, which is also
known as the chiral quark-soliton model. Given an axial-vector current with a
specified flavor, there are four different form factors of a decuplet baryon.
When we consider the singlet, triplet, and octet axial-vector currents, we have
twelve different form factors for each member of the baryon decuplet. We
compute all these axial-vector form factors of the baryon decuplet, taking into
account the rotational $1/N_c$ corrections and effects of flavor SU(3) symmetry
breaking. We find that, for a given flavor, two of the form factors for a
decuplet baryon are only independent within the present approach. We first
examine properties of the axial-vector form factors of the $\Delta^+$ isobar
and $\Omega^-$ hyperon. We also compare the results of the triplet axial-vector
form factors of $\Delta^+$ with those from lattice QCD and those of the present
work for the axial-vector constants of the baryon decuplet with the lattice
data. All the results for other members of the baryon decuplet are then
presented. The results of the axial charges are compared with those of other
works. The axial masses and axial radii are also discussed.
