TRANSLATING SOLUTIONS TO THE GAUSS CURVATURE FLOW WITH FLAT SIDES

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 Omega. This is closely is related to deriving local C-1,C-1 estimates for the degenerate Monge-Ampere equation. As a result, given a weakly convex bounded domain Omega, we establish the existence of a C-loc(1,1) translating soliton. In particular, when the boundary a partial derivative 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.