Translating Solitons to Flows by Powers of The Gaussian Curvature in Riemannian Products

We consider translating solitons to flows by positive powers $\alpha$ of the Gaussian curvature – called $K^{\alpha }$-flows – in Riemannian products $M\times \mathbb{R}\mathrm{.}$ We prove that, when $M$ is the Euclidean space $\mathbb R^n,$thesphere$\mathbb S^n,$oroneofthehyperbolicspaces$$\mathbb{H}$_{\mathbb F}^m,$ there exist complete rotational translating solitons to $K^{\alpha }$-flow in $M\times \mathbb{R}$ for certain values of $\alpha \mathrm{.}$