Uniqueness of closed self-similar solutions to the Gauss curvature flow

We show the uniqueness of strictly convex closed smooth self-similar solutions to the $\alpha$-Gauss curvature flow with $(1\mathrm{/}n)<\alpha <1+(1\mathrm{/}n)$. We introduce a Pogorelov type computation, and then we apply the strong maximum principle. Our work combined with earlier works on the Gauss Curvature flow imply that the $\alpha$-Gauss curvature flow with $(1\mathrm{/}n)<\alpha <1+(1\mathrm{/}n)$ shrinks a strictly convex closed smooth hypersurface to a round sphere.