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We prove the all-time existence of non-compact, complete, strictly convex solutions to the α-Gauss curvature flow for any positive power α.

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In this paper, we study the deformation of the 2-dimensional convex surfaces in R 3 whose speed at a point on the surface is proportional to α-power of positive part of Gauss Curvature. First, for 1 2 < α 1, we show that there is smooth solution if the initial data is smooth and strictly convex and that there is a viscosity solution with C 1,1 -estimate before the collapsing time if the initial surface is only convex. Moreover, we show that there is a waiting time effect which means the flat spot of the convex surface will persist for a while. We also show the interface between the flat side and the strictly convex side of the surface remains smooth on 0 < t < T 0 under certain necessary regularity and non-degeneracy initial conditions, where T 0 is the vanishing time of the flat side.

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