Translating solutions to the Gauss curvature flow with flat sides

Abstract

We derive local $C^2$ estimates for complete noncompact translating solitons of the Gauss curvature flow in ${ℝ}^3$ which are graphs over a convex domain $\mathrm{\Omega }$. This is closely is related to deriving local $C^{1,1}$ estimates for the degenerate Monge–Ampère equation. As a result, given a weakly convex bounded domain $\mathrm{\Omega }$, we establish the existence of a $C_{loc}^{1,1}$ translating soliton. In particular, when the boundary $\partial \mathrm{\Omega }$ has line segments, we show the existence of flat sides of the translator from a local a priori nondegeneracy estimate near the free boundary.