#### Vol. 14, No. 2, 2021

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Translating solutions to the Gauss curvature flow with flat sides

### Kyeongsu Choi, Panagiota Daskalopoulos and Ki-Ahm Lee

Vol. 14 (2021), No. 2, 595–616
##### 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 $\Omega$. 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 $\Omega$, we establish the existence of a ${C}_{loc}^{1,1}$ translating soliton. In particular, when the boundary $\partial \Omega$ has line segments, we show the existence of flat sides of the translator from a local a priori nondegeneracy estimate near the free boundary.

##### Keywords
Gauss curvature flow, translator, regularity, free boundary
Primary: 53C44
##### Milestones
Received: 20 November 2018
Revised: 9 September 2019
Accepted: 25 October 2019
Published: 20 March 2021
##### Authors
 Kyeongsu Choi Department of Mathematics Massachusetts Institute of Technology Cambridge, MA United States Korea Institute for Advanced Study Seoul South Korea Panagiota Daskalopoulos Department of Mathematics Columbia University New York, NY United States Ki-Ahm Lee Department of Mathematical Sciences Seoul National University Seoul South Korea Korea Institute for Advanced Study Seoul South Korea