Translating solutions to the Gauss curvature flow with flat sides

TL;DR

This paper establishes local regularity estimates for non-compact translating solitons of the Gauss curvature flow in three dimensions, leading to existence results for solitons with flat sides over convex domains.

Contribution

It derives new local $C^{2}$ and $C^{1,1}$ estimates for Gauss curvature flow solitons, enabling the construction of solutions with flat sides over convex domains.

Findings

Existence of $C^{1,1}_{loc}$ translating solitons over weakly convex domains.

Presence of flat sides when the boundary has line segments.

Connection between curvature flow solitons and degenerate Monge-Ampère equations.

Abstract

We derive local $C^{2}$ estimates for complete non-compact translating solitons of the Gauss curvature flow in $\mathbb{R}^3$ which are graphs over a convex domain $\Omega$. This is closely is related to deriving local $C^{1,1}$ estimates for the degenerate Monge-Amp\'ere equation. As a result, given a weakly convex bounded domain $\Omega$, we establish the existence of a $C^{1,1}_{\text{loc}}$ translating soliton. In particular, when the boundary $\partial \Omega$ has a line segment, we show the existence of flat sides of the translator from a local a'priori non-degeneracy estimate near the free-boundary.