Curvature flows with a flat side

Abstract

We study the near-the-interface behavior of a compact convex scalar curvature flow with a flat side. Under suitable initial conditions on the flat side, we show that the interface propagates with a finite and non-degenerate speed and the level set with a finite speed, until the flat side vanishes. Then we get optimal derivative estimates of the pressure-like function, non-degeneracy of the speed of the level set, optimal decay estimates of curvatures near the interface, and a generalized version of Kim-Lee-Rhees curvature lower bound, from which we obtain the Hölder regularity of the ratio of the curvature to the optimal decay rate up to the free boundary. In the end, we demonstrate the all-time existence of a solution which is smooth up to the interface in its support.