Long time regularity of the p-Gauss curvature flow with flat side

In this paper, we prove the long time regularity of the interface in the p-Gauss curvature flow with flat side in all dimensions for p>1n. Here the interface is the boundary of the flat part in the flow. In dimension $2$, this problem was solved in {DL2004} for $p=1$ and in {KimLeeRhee2013} for p∈(1/2,1). We utilize the duality method to transform the Gauss curvature flow to a singular parabolic Monge-Amp\`ere equation, and prove the regularity of the interface by studying the asymptotic cone of the parabolic Monge-Amp\`ere equation in the polar coordinates.