Translating surfaces under flows by sub-affine-critical powers of Gauss curvature

We classify the surfaces translating under the flows by sub-affine-critical powers of the Gauss curvature. This, in particular, lists all translating solitons possibly model Type II singularities for convex closed solutions in all positive powers. The surfaces are entire graphs, and therefore our result corresponds to the Liouville theorem for the degenerate Monge--Amp\`ere equations $\det D^2 u=(1+|Du|^2)^{2-\frac{1}{2\alpha}}$ on $\mathbb{R}^2$ in the range $0<\alpha <1/4$. The result also reveals that the moduli spaces of solutions are homeomorphic to either Euclidean spaces or cylinders.