  Effect of exchange anisotropy on the relaxation time of spin and vector
chirality is studied for the antiferromagnetic classical Heisenberg model on
the triangular lattice by using the nonequilibrium relaxation Monte Carlo
method. We identify the Berezinskii-Kosterlitz-Thouless (BKT) transition and
the chiral transition in a wide range of the anisotropy, even for very small
anisotropy of $\sim$ 0.01%. As the anisotropy decreases, both the critical
temperatures steeply decrease, while the BKT critical region becomes
divergently wide. We elucidate a sharp "V shape" of the phase diagram around
the isotropic Heisenberg point, which suggests that the isotropic case is
exceptionally singular and the associated Z vortex transition will be isolated
from the BKT and chiral transitions. We discuss the relevance of our results to
peculiar behavior of the spin relaxation time observed experimentally in
triangular antiferromagnets.
