The domain growth after a quench to very low, finite temperatures is analyzed by scaling theory and Monte Carlo simulation. The growth exponent for the excess energy ΔE(t)∼ t − n is found to be n∼(1/4. The scaling theory gives exactly n=(1/4 for cases of hierarchical movement of domain walls. This explains the existence of a slow growth universality class. It is shown to be a singular Allen-Cahn class, to which belongs systems with domain walls of both exactly zero and finite curvature. The model studied has continuous variables, nonconserved order parameter, and has two kinds of domain walls: sharp, straight, stacking faults and broad, curved, solitonlike walls.