On existence of hypersurfaces translating by powers of Gauss curvature

In this paper we construct complete convex hypersurfaces in ${\mathbb{R}}^{n+1}$ which translate under the flow by powers $\alpha \in (0,\frac{1}{n+2})$ of the Gauss curvature. The level set of each solution is asymptotic to a shrinking soliton for the flow by power $\frac{\alpha }{1-\alpha }$ of the Gauss curvature in ${\mathbb{R}}^n$. For example, our construction reveals the existence of translators whose level set converges to the sphere, simplex, hypercube and so on. The translating solitons exist as a family whose parameters correspond to Jacobi fields, solutions to linearized equation around the asymptotic profile.