  This paper shows the existence of convex translating surfaces under the flow
by the $\alpha$-th power of Gauss curvature for the sub-affine-critical regime
$ 0 < \alpha < 1/4$. The key aspect of our study is that our ansatz at infinity
is the graph of homogeneous functions whose level sets are closed curves
shrinking under the flow by the $\frac{\alpha}{1-\alpha}$-th power of
curvature. For each ansatz, we construct a family of translating surfaces
generated by the Jacobi fields with effective growth rates. Moreover, the
construction shows quantitative estimate on the rate of convergence between
different translators to each other, which is required to show the continuity
of the family. As a result, the family is regarded as a topological manifold.
The construction in this paper will become the ground of forthcoming research,
where we aim to prove that every translating surface must correspond to one of
the solutions obtained herein, classifying translating surfaces and identifying
the topology of the moduli space.
