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

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