Free-boundary regularity on the focusing problem for the Gauss Curvature Flow with flat sides

Abstract

We consider the motion of a compact weakly convex two-dimensional surface of revolution \(\Sigma\) under the Gauss Curvature Flow. We assume that the initial surface has a flat side and as a consequence the parabolic equation describing the motion of the hypersurface becomes degenerate at points where the curvature is zero. Expressing the strictly convex part of the surface near the interface as the graph of a function \(z=f(r,t)\), we show that if at ti me \(t=0, g=\sqrt f\) vanishes linearly at the flat side, then \(g(r,t)\) will become smooth up to the interface for $t >0$ and it will remain smooth up to the focusing time T of the flat side. We also show that at the focusing time of the flat side, the function g is of class \(C^{1,\beta}\) for all \(\beta < \) and no better than \(C^{1,{ 2 \over 5}}\). This implies that at the focusing time the surface \(\Sigma\) is of class \(C^{2,\beta}\) for all $\beta < $ and no better than \(C^{2,{2 \over 5}}\). In the case of the evolution Monge-Ampere equation, we find the exact self-similar profile of the function g at its focusing time.